Problem p02634 (C++) — buggy submission, verdict: Runtime Error

/*temp*/
//
//
//
//
// #undef _DEBUG
// #pragma GCC optimize("Ofast")
// 不動小数点の計算高速化
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
// #include <boost/multiprecision/cpp_int.hpp>

#ifdef _DEBUG
#include "template.h"
#else
#if __cplusplus >= 201703L
/*Atcoderでしか使えない(c++17 && このテンプレートが使えるならAtcoder)*/
#include <boost/sort/pdqsort/pdqsort.hpp>
#define fast_sort boost::sort::pdqsort
#endif
#endif

#ifndef _DEBUG
#ifndef UNTITLED15_TEMPLATE_H
#define UNTITLED15_TEMPLATE_H

#ifdef _DEBUG
#include "bits_stdc++.h"
#else
#include <bits/stdc++.h>
#endif

#ifndef fast_sort
#define fast_sort sort
#endif

// #define use_pq
#define use_for
#define use_for_each
#define use_sort
#define use_fill
#define use_rand
#define use_mgr
#define use_rui
#define use_compress
//
//
//
//
//
//
#define use_pbds
#ifdef use_pbds
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/tree_policy.hpp>

using namespace std;
using namespace std::chrono;

template <class T, class U>
void operator+=(__gnu_pbds::priority_queue<T, greater<T>,
                                           __gnu_pbds::rc_binomial_heap_tag> &a,
                const U &v) {
  a.push(v);
}
template <class T, class U>
void operator+=(
    __gnu_pbds::priority_queue<T, less<T>, __gnu_pbds::rc_binomial_heap_tag> &a,
    const U &v) {
  a.push(v);
}

template <class T, class U, class W, class X>
auto count(__gnu_pbds::gp_hash_table<T, U, W> &a, X k) {
  return a.find(k) != a.end();
}
#endif

/*@formatter:off*/
#define ll long long
using sig_dou = double;

// マクロ省略形 関数等
#define arsz(a) (sizeof(a) / sizeof(a[0]))
#define sz(a) ((ll)(a).size())
#define mp make_pair
#define mt make_tuple
#define pb pop_back
#define pf push_front
#define eb emplace_back
#define ALL(a) (a).begin(), (a).end()
#define RALL(a) (a).rbegin(), (a).rend()

template <class T, class U> auto max(T a, U b) { return a > b ? a : b; }
template <class T, class U> auto min(T a, U b) { return a < b ? a : b; }
// optional<T>について下でオーバーロード(nullopt_tを左辺右辺について単位元として扱う)
template <class T, class U> bool chma(T &a, const U &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <class T, class U> bool chmi(T &a, const U &b) {
  if (b < a) {
    a = b;
    return true;
  }
  return false;
}

// メタ系 meta
template <typename T, typename U = typename T::value_type>
std::true_type value_type_tester(signed);
template <typename T> std::false_type value_type_tester(long);

template <typename T>
struct has_value_type : decltype(value_type_tester<T>(0)) {};

// A<T>でTを返す
template <class T> using decl_t = typename T::value_type;
// A<B<.....T>>でTを返す
template <class T, bool end = !has_value_type<T>::value> struct decl2_ {
  typedef T type;
};
template <class T> struct decl2_<T, false> {
  typedef typename decl2_<decl_t<T>>::type type;
};
template <class T> using decl2_t = typename decl2_<T>::type;
// #define decl_max(a, b) decltype(max(MAX<decltype(a)>(), MAX<decltype(b)>()))
#define is_same2(T, U) is_same<T, U>::value
template <class T> struct is_vector : std::false_type {};
template <class T> struct is_vector<std::vector<T>> : std::true_type {};
// 大きい型を返す max_type<int, char>::type
// todo mintがlong long より小さいと判定されるためバグる
template <class T1, class T2, bool t1_bigger = (sizeof(T1) > sizeof(T2))>
struct max_type {
  typedef T1 type;
};
template <class T1, class T2> struct max_type<T1, T2, false> {
  typedef T2 type;
};

template <class T> struct vec_rank : integral_constant<int, 0> {};
template <class T>
struct vec_rank<vector<T>> : integral_constant<int, vec_rank<T>{} + 1> {};

// N個のTを並べたtupleを返す
// tuple_n<3, int>::type  tuple<int, int, int>
template <size_t N, class T, class... Arg> struct tuple_n {
  typedef typename tuple_n<N - 1, T, T, Arg...>::type type;
};
template <class T, class... Arg> struct tuple_n<0, T, Arg...> {
  typedef tuple<Arg...> type;
};

struct dummy_t1 {};
struct dummy_t2 {};
struct dummy_t3 {};
struct dummy_t4 {};
struct dummy_t5 {};
struct dummy_t6 {};
// template<class T, require(is_integral<T>::value)>など
#define require_t(bo) enable_if_t<bo> * = nullptr

// 複数でオーバーロードする場合、引数が同じだとうまくいかないため
// require_arg(bool, dummy_t1)
// require_arg(bool, dummy_t2)等とする
#define require_arg1(bo) enable_if_t<bo> * = nullptr
#define require_arg2(bo, dummy_type) enable_if_t<bo, dummy_type> * = nullptr
#define require_arg(...)                                                       \\
  over2(__VA_ARGS__, require_arg2, require_arg1)(__VA_ARGS__)
//->//enable_if_tのtを書き忘れそうだから
#define require_ret(bo, ret_type) enable_if_t<bo, ret_type>

#define int long long // todo 消したら動かない intの代わりにsignedを使う
auto start_time = system_clock::now();
auto past_time = system_clock::now();

#define debugName(VariableName) #VariableName
// 最大引数がN
#define over2(o1, o2, name, ...) name
#define over3(o1, o2, o3, name, ...) name
#define over4(o1, o2, o3, o4, name, ...) name
#define over5(o1, o2, o3, o4, o5, name, ...) name
#define over6(o1, o2, o3, o4, o5, o6, name, ...) name
#define over7(o1, o2, o3, o4, o5, o6, o7, name, ...) name
#define over8(o1, o2, o3, o4, o5, o6, o7, o8, name, ...) name
#define over9(o1, o2, o3, o4, o5, o6, o7, o8, o9, name, ...) name
#define over10(o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, name, ...) name

#ifdef _DEBUG
string message;
string res_mes;
bool was_deb = false;
[[noreturn]] void my_exit(signed e = 0) {
  if (was_deb && sz(res_mes)) {
    cerr << "result = " << endl << res_mes << endl;
  }
  if (sz(message)) {
    cerr << "****************************" << endl;
    cerr << "Note." << endl;
    cerr << message << endl;
    cerr << "****************************" << endl;
  }
  exit(e);
}
#define exit my_exit

#endif
void assert2(bool b, const string &s = "") {
  if (!b) {
    cerr << s << endl;
    exit(1); /*assert(0);*/
  }
}

// my_nulloptをあらゆる操作の単位元的な物として扱う
// vectorの参照外時に返したり、右辺値として渡されたときに何もしないなど
struct my_nullopt_t {
} my_nullopt;
#define nullopt_t my_nullopt_t
#define nullopt my_nullopt
/*@formatter:off*/
// 値が無いときは、setを使わない限り代入できない
//=を使っても無視される
template <class T> struct my_optional {
private:
  bool is_null;
  T v;

public:
  typedef T value_type;
  my_optional() : is_null(true) {}
  my_optional(const nullopt_t &) : is_null(true) {}
  my_optional(const T &v) : v(v), is_null(false) {}
  bool has_value() const { return !is_null; }
  T &value() {
    static string mes = "optional has no value";
    assert2(!is_null, mes);
    return v;
  }
  const T &value() const {
    static string mes = "optional has no value";
    assert2(!is_null, mes);
    return v;
  }
  void set(const T &nv) {
    is_null = false;
    v = nv;
  }
  template <class U> void operator=(const U &v) {
    set(v); // null状態でも代入出来るようにした
    //        if (has_value())value() = v; else return;
  }
  template <class U> void operator=(const my_optional<U> &v) {
    if (/*has_value() && */ v.has_value())
      (*this) = v;
    else
      return;
  }
  /*@formatter:off*/
  void reset() { is_null = true; }
  void operator=(const nullopt_t &) { reset(); }
  template <require_t(!is_same2(T, bool))> explicit operator bool() {
    return !is_null;
  }
  // nullの時はエラー
  operator T &() { return value(); }
  operator const T &() const { return value(); }
  my_optional<T> operator++() {
    if (this->has_value()) {
      this->value()++;
      return *this;
    } else {
      return *this;
    }
  }
  my_optional<T> operator++(signed) {
    if (this->has_value()) {
      auto tem = *this;
      this->value()++;
      return tem;
    } else {
      return *this;
    }
  }
  my_optional<T> operator--() {
    if (this->has_value()) {
      this->value()--;
      return *this;
    } else {
      return *this;
    }
  }
  my_optional<T> operator--(signed) {
    if (this->has_value()) {
      auto tem = *this;
      this->value()--;
      return tem;
    } else {
      return *this;
    }
  }
};
template <class T> istream &operator>>(istream &iss, my_optional<T> &v) {
  T val;
  iss >> val;
  v.set(val);
  return iss;
}

#define optional my_optional
template <class T> using opt = my_optional<T>;
// template<class T, class A = std::allocator<T>> struct debtor : std::vector<T,
// A> {
template <class Key, class T, class Compare = less<Key>,
          class Allocator = allocator<pair<const Key, T>>>
struct o_map : std::map<Key, optional<T>, Compare, Allocator> {
  optional<T> emp;
  o_map() : std::map<Key, optional<T>, Compare, Allocator>() {}
  auto operator()(const nullopt_t &) { return nullopt; }
  optional<T> &operator()(const optional<Key> &k) {
    if (k.has_value()) {
      return std::map<Key, optional<T>, Compare, Allocator>::operator[](
          k.value());
    } else {
      emp.reset();
      return emp;
    }
  }
  optional<T> &operator()(const Key &k) {
    auto &v = std::map<Key, optional<T>, Compare, Allocator>::operator[](k);
    if (v.has_value())
      return v;
    else {
      v.set(0);
      return v;
    }
  }
  template <class U> void operator[](U) {
    static string mes = "s_map cant []";
    assert2(0, mes);
  }
};
// 以下、空のoptionalをnulloptと書く
// ov[-1(参照外)] でnulloptを返す
// ov[nullopt] で nulloptをかえす

template <class T> struct ov {
  optional<T> emp;
  vector<optional<T>> v;
  ov(int i = 0, T val = 0) : v(i, val) {}
  template <class U> ov(const U &rhs) {
    v.resize(sz(rhs));
    for (int i = 0; i < sz(rhs); i++)
      v[i].set(rhs[i]);
  }
  optional<T> &operator()(int i) {
    if (i < 0 || sz(v) <= i) {
      emp.reset();
      return emp;
    } else {
      return v[i];
    }
  }
  optional<T> &operator()(const nullopt_t &) { return operator()(-1); }
  optional<T> &operator()(const optional<T> &i) {
    if (i.has_value())
      return operator()(i.value());
    else {
      return operator()(-1);
    }
  }
  /*@formatter:off*/
};
template <class T> string deb_tos(const ov<T> &v) { return deb_tos(v.v); }

// vectorに対しての処理は.vを呼ぶ
template <class T> class ovv {
  optional<T> emp;

public:
  vector<vector<optional<T>>> v;
  ovv(int i = 0, int j = 0, T val = 0) : v(i, vector<optional<T>>(j, val)) {}
  ovv(const vector<vector<T>> &S) {
    v.resize(S.size());
    for (int h = 0; h < sz(S); h++) {
      v[h].resize(sz(S[h]));
      for (int w = 0; w < sz(S[h]); w++) {
        v[h][w].set(S[h][w]);
      }
    }
  }
  optional<T> &operator()(int i, int j) {
    if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {
      emp.reset();
      return emp;
    } else {
      return v[i][j];
    }
  }
  // 再帰ver 遅いと思う
//    optional<T>& gets(optional<T>& v){return v;}
//    template<class V, class H, class... U> optional<T>& gets(V& v, H i, U...
//    tail){        if constexpr(is_same2(H, nullopt_t))return
//    operator()(-1,-1);        else if constexpr(is_same2(H, optional<int>)){
//    if(i.has_value())return gets(v[(int)i], tail...);            else return
//    operator()(-1,-1);        }else if constexpr(is_integral<H>::value){
//    return gets(v[(int)i], tail...);        }else{            assert(0);
//    return emp;        }    }
#if __cplusplus >= 201703L
  // if constexprバージョン 上が遅かったらこれで
  template <class U, class V>
  optional<T> &operator()(const U &i, const V &j) { /*駄目な場合を除外*/
    if constexpr (is_same2(U, nullopt_t) || is_same2(U, nullopt_t)) {
      return operator()(-1, -1); /* o, o*/
    } else if constexpr (is_same2(U, optional<int>) &&
                         is_same2(V, optional<int>)) {
      return operator()(i.has_value() ? (int)i : -1,
                        j.has_value() ? (int)j : -1); /* o, x*/
    } else if constexpr (is_same2(U, optional<int>)) {
      return operator()(i.has_value() ? (int)i : -1, (int)j); /* x, o*/
    } else if constexpr (is_same2(V, optional<int>)) {
      return operator()((int)i, j.has_value() ? (int)j : -1); /* x, x*/
    } else {
      return operator()((int)i, (int)j);
    }
  }
#endif
  operator const vector<vector<optional<T>>> &() { return v; }
};
template <class T> istream &operator>>(istream &iss, ovv<T> &a) {
  for (int h = 0; h < sz(a); h++) {
    for (int w = 0; w < sz(a[h]); w++) {
      iss >> a.v[h][w];
    }
  }
  return iss;
}
template <class T> ostream &operator<<(ostream &os, ovv<T> &a) {
  os << a.v;
  return os;
}
template <class T> string deb_tos(const ovv<T> &v) { return deb_tos(v.v); }
template <class T> struct ov3 {
  optional<T> emp;
  vector<vector<vector<optional<T>>>> v;
  ov3(int i, int j, int k, T val = 0)
      : v(i, vector<vector<optional<T>>>(j, vector<optional<T>>(k, val))) {}
  optional<T> &operator()(int i, int j, int k) {
    if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {
      if (k < 0 || sz(v[i][j]) <= k) {
        emp.reset();
        return emp;
      }
    }
    return v[i][j][k];
  }

private:
#if __cplusplus >= 201703L
  // 再帰ver 遅いと思う
  template <class V, class H> optional<T> &gets(V &nowv, H i) {
    if constexpr (is_same2(H, nullopt_t)) {
      emp.reset();
      return emp;
    } else if constexpr (is_same2(H, optional<int>)) {
      if (i.has_value()) {
        return nowv[(int)i];
      } else {
        emp.reset();
        return emp;
      }
    } else if constexpr (is_integral<H>::value) {
      return nowv[(int)i];
    } else {
      static string mes = "ov3 error not index";
      assert2(0, mes);
      emp.reset();
      return emp;
    }
  }
  // todo const &消した
  template <class V, class H, class... U>
  optional<T> &gets(V &nowv, H i, U... tail) {
    if constexpr (is_same2(H, nullopt_t)) {
      emp.reset();
      return emp;
    } else if constexpr (is_same2(H, optional<int>)) {
      if (i.has_value()) {
        return gets(nowv[(int)i], tail...);
      } else {
        emp.reset();
        return emp;
      }
    } else if constexpr (is_integral<H>::value) {
      return gets(nowv[(int)i], tail...);
    } else {
      static string mes = "ov3 error not index";
      assert2(0, mes);
      emp.reset();
      return emp;
    }
  }
#endif
public:
  template <class U, class V, class W> optional<T> &operator()(U i, V j, W k) {
    return gets(v, i, j, k);
  }
  /*@formatter:off*/
};
template <class T> string deb_tos(const ov3<T> &v) { return deb_tos(v.v); }
// nullopt_t
// 優先順位
// null, [opt, tem]

// + と += は違う意味を持つ
// val+=null : val
// val+null  : null
//
//+は途中計算
//+=は最終的に格納したい値にだけ持たせる
//+=がvoidを返すのは、途中計算で使うのを抑制するため

// nulloptを考慮する際、計算途中では+を使ってnulloptを作り
// 格納する際は+=で無効にする必要がある

// 演算子==
// optional<int>(10) == 10

// 全ての型に対応させ、value_typeが等しいかを見るようにするのもありかも

// null同士を比較する状況はおかしいのではないか
bool operator==(const nullopt_t &, const nullopt_t &) {
  assert2(0, "nul == null  cant hikaku");
  return false;
}
template <class T> bool operator==(const nullopt_t &, const T &) {
  return false;
}
template <class T> bool operator!=(const nullopt_t &, const T &) {
  return true;
}
template <class T> bool operator==(const T &, const nullopt_t &) {
  return false;
}
template <class T> bool operator!=(const T &, const nullopt_t &) {
  return true;
}
// nullを

nullopt_t &operator+(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator-(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator*(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator/(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator+=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator-=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator*=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator/=(const nullopt_t &, const nullopt_t &) { return nullopt; }

template <class ANY> nullopt_t operator+(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator-(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator*(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator/(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator+(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator-(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator*(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator/(const ANY &, const nullopt_t &) {
  return nullopt;
}

template <class ANY> void operator+=(nullopt_t &, const ANY &) {}
template <class ANY> void operator-=(nullopt_t &, const ANY &) {}
template <class ANY> void operator*=(nullopt_t &, const ANY &) {}
template <class ANY> void operator/=(nullopt_t &, const ANY &) {}
template <class ANY> void operator+=(ANY &, const nullopt_t &) {}
template <class ANY> void operator-=(ANY &, const nullopt_t &) {}
template <class ANY> void operator*=(ANY &, const nullopt_t &) {}
template <class ANY> void operator/=(ANY &, const nullopt_t &) {}

template <class T> struct is_optional : false_type {};
template <class T> struct is_optional<optional<T>> : true_type {};
template <class T, class U>
true_type both_optional(optional<T> t, optional<U> u);
false_type both_optional(...);

template <class T, class U>
class opt_check : public decltype(both_optional(declval<T>(), declval<U>())) {};

// optionalは同じ型同士しか足せない
//(o, t), (t, o), (o, o)
#define opt_tem(op)                                                            \\
  template <class O, class O_ret = decltype(declval<O>() op declval<O>())>     \\
  optional<O_ret> operator op(const optional<O> &opt1,                         \\
                              const optional<O> &opt2) {                       \\
    if (!opt1.has_value() || !opt2.has_value()) {                              \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt1.value() op opt2.value());                    \\
    }                                                                          \\
  }                                                                            \\
  template <class O, class T,                                                  \\
            class O_ret = decltype(declval<O>() op declval<O>())>              \\
  auto operator op(const optional<O> &opt, const T &tem)                       \\
      ->require_ret(!(opt_check<optional<O>, T>::value), optional<O_ret>) {    \\
    if (!opt.has_value()) {                                                    \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt.value() op tem);                              \\
    }                                                                          \\
  }                                                                            \\
  template <class O, class T,                                                  \\
            class O_ret = decltype(declval<O>() op declval<O>())>              \\
  auto operator op(const T &tem, const optional<O> &opt)                       \\
      ->require_ret(!(opt_check<optional<O>, T>::value), optional<O_ret>) {    \\
    if (!opt.has_value()) {                                                    \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt.value() op tem);                              \\
    }                                                                          \\
  }
/*@formatter:off*/
opt_tem(+) opt_tem(-) opt_tem(*) opt_tem(/)
    // 比較はoptional<bool>を返す
    opt_tem(<) opt_tem(>) opt_tem(<=) opt_tem(>=)

    /*@formatter:on*/ /*@formatter:off*/

    template <class O, class T>
    bool operator==(const optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    return opt.value() == tem;
  } else
    return nullopt == tem;
}
template <class O, class T>
bool operator!=(const optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    return opt.value() != tem;
  } else
    return nullopt != tem;
}
template <class O, class T>
bool operator==(const T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    return opt.value() == tem;
  } else
    return nullopt == tem;
}
template <class O, class T>
bool operator!=(const T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    return opt.value() != tem;
  } else
    return nullopt != tem;
}

template <class O>
bool operator==(const optional<O> &opt1, const optional<O> &opt2) {
  if (opt1.has_value() != opt2.has_value()) {
    return false;
  } else if (opt1.has_value()) {
    return opt1.value() == opt2.value();
  } else {
    return nullopt == nullopt;
  }
}
template <class O>
bool operator!=(const optional<O> &opt1, const optional<O> &opt2) {
  return !(opt1 == opt2);
}

//(a+=null) != (a=a+null)
// a            null
template <class T, class O> void operator+=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem += opt.value();
  }
}
template <class T, class O> void operator-=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem -= opt.value();
  }
}
template <class T, class O> void operator*=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem *= opt.value();
  }
}
template <class T, class O> void operator/=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem /= opt.value();
  }
}

template <class T, class O> void operator+=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() += tem;
  }
}
template <class T, class O> void operator-=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() -= tem;
  }
}
template <class T, class O> void operator*=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() *= tem;
  }
}
template <class T, class O> void operator/=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() /= tem;
  }
}
//
template <class Ol, class Or>
void operator+=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl += opr.value();
  }
}
template <class Ol, class Or>
void operator-=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl -= opr.value();
  }
}
template <class Ol, class Or>
void operator*=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl *= opr.value();
  }
}
template <class Ol, class Or>
void operator/=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl /= opr.value();
  }
}

/*@formatter:off*/

template <class U> auto max(const nullopt_t &, const U &val) { return val; }
template <class U> auto max(const U &val, const nullopt_t &) { return val; }
template <class U> auto min(const nullopt_t &, const U &val) { return val; }
template <class U> auto min(const U &val, const nullopt_t &) { return val; }

template <class T, class U> auto max(const optional<T> &opt, const U &val) {
  if (opt.has_value())
    return max(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto max(const U &val, const optional<T> &opt) {
  if (opt.has_value())
    return max(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto min(const optional<T> &opt, const U &val) {
  if (opt.has_value())
    return min(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto min(const U &val, const optional<T> &opt) {
  if (opt.has_value())
    return min(opt.value(), val);
  else
    return val;
}

// null , optional, T
bool chma(nullopt_t &, const nullopt_t &) { return false; }
template <class T> bool chma(T &opt, const nullopt_t &) { return false; }
template <class T> bool chma(nullopt_t &, const T &opt) { return false; }
template <class T> bool chma(optional<T> &olv, const optional<T> &orv) {
  if (orv.has_value()) {
    return chma(olv, orv.value());
  } else
    return false;
}
template <class T, class U> bool chma(optional<T> &opt, const U &rhs) {
  if (opt.has_value()) {
    return chma(opt.value(), rhs);
  } else
    return false;
}
template <class T, class U> bool chma(T &lhs, const optional<U> &opt) {
  if (opt.has_value()) {
    return chma(lhs, opt.value());
  } else
    return false;
}

bool chmi(nullopt_t &, const nullopt_t &) { return false; }
template <class T> bool chmi(T &opt, const nullopt_t &) { return false; }
template <class T> bool chmi(nullopt_t &, const T &opt) { return false; }
template <class T> bool chmi(optional<T> &olv, const optional<T> &orv) {
  if (orv.has_value()) {
    return chmi(olv, orv.value());
  } else
    return false;
}
template <class T, class U> bool chmi(optional<T> &opt, const U &rhs) {
  if (opt.has_value()) {
    return chmi(opt.value(), rhs);
  } else
    return false;
}
template <class T, class U> bool chmi(T &lhs, const optional<U> &opt) {
  if (opt.has_value()) {
    return chmi(lhs, opt.value());
  } else
    return false;
}

template <class T> ostream &operator<<(ostream &os, optional<T> p) {
  if (p.has_value())
    os << p.value();
  else
    os << "e";
  return os;
}
template <class T> using opt = my_optional<T>;
struct xorshift {
  /*@formatter:on*/
  static uint64_t splitmix64(uint64_t x) {
    x += 0x9e3779b97f4a7c15;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
  }
  /*@formatter:off*/
  size_t operator()(const uint64_t &x) const {
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    return splitmix64(x + FIXED_RANDOM);
  }
  size_t operator()(const std::pair<ll, ll> &x) const {
    ll v = ((x.first) << 32) | x.second;
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    return splitmix64(v + FIXED_RANDOM);
  }
  template <class T, class U>
  size_t operator()(const std::pair<T, U> &x) const {
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    uint64_t hasx = splitmix64(x.first);
    uint64_t hasy = splitmix64(x.second + FIXED_RANDOM);
    return hasx ^ hasy;
  }
  template <class T> size_t operator()(const vector<T> &x) const {
    uint64_t has = 0;
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    uint64_t rv = splitmix64(FIXED_RANDOM);
    for (int i = 0; i < sz(x); i++) {
      uint64_t v = splitmix64(x[i] + rv);
      has ^= v;
      rv = splitmix64(rv);
    }
    return has;
  }
};

#ifdef _DEBUG
// #define use_debtor
// template<class T, class U, class X> auto count(unordered_map<T, U> &a, X k) {
// return a.find(k) != a.end(); }

#ifdef use_debtor
// https://marycore.jp/prog/cpp/class-extension-methods/ 違うかも
template <class T, class A = std::allocator<T>>
struct debtor : std::vector<T, A> {
  using std::vector<T, A>::vector;
  template <class U> int deb_v(U a, int v) { return v; }
  template <class U> int deb_v(debtor<U> &a, int v = 0) {
    cerr << a.size() << " ";
    return deb_v(a.at(0), v + 1);
  }
  template <class U> void deb_o(U a) { cerr << a << " "; }
  template <class U> void deb_o(debtor<U> &a) {
    for (int i = 0; i < min((int)a.size(), 15ll); i++) {
      deb_o(a[i]);
    }
    if ((int)a.size() > 15) {
      cerr << "...";
    }
    cerr << endl;
  }
  typename std::vector<T>::reference my_at(typename std::vector<T>::size_type n,
                                           vector<int> &ind) {
    if (n < 0 || n >= (int)this->size()) {
      int siz = (int)this->size();
      cerr << "vector size = ";
      int dim = deb_v((*this));
      cerr << endl;
      ind.push_back(n);
      cerr << "out index at  ";
      for (auto &&i : ind) {
        cerr << i << " ";
      }
      cerr << endl;
      cerr << endl;
      if (dim <= 2) {
        deb_o((*this));
      }
      exit(0);
    }
    return this->at(n);
  }
  typename std::vector<T>::reference
  operator[](typename std::vector<T>::size_type n) {
    if (n < 0 || n >= (int)this->size()) {
      int siz = (int)this->size();
      cerr << "vector size = ";
      int dim = deb_v((*this));
      cerr << endl;
      cerr << "out index at  " << n << endl;
      cerr << endl;
      if (dim <= 2) {
        deb_o((*this));
      }
      exit(0);
    }
    return this->at(n);
  }
};
#define vector debtor
#endif
#ifdef use_pbds
template <class T> struct my_pbds_tree {
  set<T> s;
  auto begin() { return s.begin(); }
  auto end() { return s.end(); }
  auto rbegin() { return s.rbegin(); }
  auto rend() { return s.rend(); }
  auto empty() { return s.empty(); }
  auto size() { return s.size(); }
  void clear() { s.clear(); }
  template <class U> void insert(U v) { s.insert(v); }
  template <class U> void operator+=(U v) { insert(v); }
  template <class F> auto erase(F v) { return s.erase(v); }
  template <class U> auto find(U v) { return s.find(v); }
  template <class U> auto lower_bound(U v) { return s.lower_bound(v); }
  template <class U> auto upper_bound(U v) { return s.upper_bound(v); }
  auto find_by_order(ll k) {
    auto it = s.begin();
    for (ll i = 0; i < k; i++)
      it++;
    return it;
  }
  auto order_of_key(ll v) {
    auto it = s.begin();
    ll i = 0;
    for (; it != s.end() && *it < v; i++)
      it++;
    return i;
  }
};
#define pbds(T) my_pbds_tree<T>
#endif

// 区間削除は出来ない
// gp_hash_tableでcountを使えないようにするため
template <class K, class V> using umap_f = unordered_map<K, V>;
template <typename K, typename T> using map_f = map<K, T>;

#else
#define endl '\\n'
// umapはunorderd_mapになる
// umapiはgp_hash_table

// find_by_order(k) k番目のイテレーター
// order_of_key(k)  k以上が前から何番目か
#define pbds(U)                                                                \\
  __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, \\
                   __gnu_pbds::tree_order_statistics_node_update>

template <class K, class V>
struct umap_f : public __gnu_pbds::gp_hash_table<K, V, xorshift> {
  int count(const K &k) { return this->find(k) != this->end(); }
};
template <typename K, typename T>
using map_f = __gnu_pbds::tree<K, T, less<K>, __gnu_pbds::rb_tree_tag,
                               __gnu_pbds::tree_order_statistics_node_update>;

#endif
#define umapi unordered_map<ll, ll>
#define umapp unordered_map<P, ll>
#define umappp unordered_map<P, P>
#define umapu unordered_map<uint64_t, ll>
#define umapip unordered_map<ll, P>

template <class T, class U, class X> auto count(unordered_map<T, U> &a, X k) {
  return a.find(k) != a.end();
}
/*@formatter:off*/
#ifdef use_pbds
template <class U, class L>
void operator+=(
    __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag,
                     __gnu_pbds::tree_order_statistics_node_update> &s,
    L v) {
  s.insert(v);
}
#endif
// 衝突対策
#define ws ws_

typedef tuple<ll, ll, ll> T;
typedef tuple<ll, ll, ll, ll> F;
// T mt(ll a, ll b, ll c) { return T(a, b, c); }
// F mf(ll a, ll b, ll c, ll d) { return F(a, b, c, d); }

// 関数内をまとめる
// 初期値l=-1, r=-1
void set_lr12(int &l, int &r, int n) { /*r==-1*/
  if (r == -1) {
    if (l == -1) {
      l = 0;
      r = n;
    } else {
      r = l;
      l = 0;
    }
  }
}
//@マクロ省略系 型,構造
// using で元のdoubleを同時に使えるはず
#define double_big

#ifdef double_big
#define double long double
// #define pow powl
#endif
using dou = double;

/*@formatter:off*/
template <class T> T MAX() { return numeric_limits<T>::max(); }
template <class T> T MIN() { return numeric_limits<T>::min(); }
constexpr ll inf = (ll)1e9 + 100;
constexpr ll linf = (ll)1e18 + 100;
constexpr dou dinf = (dou)linf * linf;
constexpr char infc = '{';
const string infs = "{";

template <class T> T INF() { return MAX<T>() / 2; }
template <> signed INF() { return inf; }
template <> ll INF() { return linf; }
template <> double INF() { return dinf; }
template <> char INF() { return infc; }
template <> string INF() { return infs; }

const double eps = 1e-9;

// #define use_epsdou
#ifdef use_epsdou
// 基本コメントアウト
struct epsdou {
  double v;
  epsdou(double v = 0) : v(v) {}
  template <class T> epsdou &operator+=(T b) {
    v += (double)b;
    return (*this);
  }
  template <class T> epsdou &operator-=(T b) {
    v -= (double)b;
    return (*this);
  }
  template <class T> epsdou &operator*=(T b) {
    v *= (double)b;
    return (*this);
  }
  template <class T> epsdou &operator/=(T b) {
    v /= (double)b;
    return (*this);
  }
  epsdou operator+(epsdou b) { return v + (double)b; }
  epsdou operator-(epsdou b) { return v - (double)b; }
  epsdou operator*(epsdou b) { return v * (double)b; }
  epsdou operator/(epsdou b) { return v / (double)b; }
  epsdou operator-() const { return epsdou(-v); }
  template <class T> bool operator<(T b) { return v < (double)b; }
  template <class T> bool operator>(T b) {
    auto r = (double)b;
    return v > (double)b;
  }
  template <class T> bool operator==(T b) { return fabs(v - (double)b) <= eps; }
  template <class T> bool operator<=(T b) {
    return v < (double)b || fabs(v - b) <= eps;
  }
  template <class T> bool operator>=(T b) {
    return v > (double)b || fabs(v - b) <= eps;
  }
  operator double() { return v; }
};
template <> epsdou MAX() { return MAX<double>(); }
template <> epsdou MIN() { return MIN<double>(); }
// priqrity_queue等で使うのに必要
bool operator<(const epsdou &a, const epsdou &b) { return a.v < b.v; }
bool operator>(const epsdou &a, const epsdou &b) { return a.v > b.v; }
istream &operator>>(istream &iss, epsdou &a) {
  iss >> a.v;
  return iss;
}
ostream &operator<<(ostream &os, epsdou &a) {
  os << a.v;
  return os;
}
#define eps_conr_t(o)                                                          \\
  template <class T> epsdou operator o(T a, epsdou b) { return (dou)a o b.v; }
#define eps_conl_t(o)                                                          \\
  template <class T> epsdou operator o(epsdou a, T b) { return a.v o(dou) b; }
eps_conl_t(+) eps_conl_t(-) eps_conl_t(*) eps_conl_t(/) eps_conr_t(+)
    eps_conr_t(-) eps_conr_t(*) eps_conr_t(/)
// template<class U> epsdou max(epsdou a, U b){return a.v>b ? a.v: b;}
// template<class U> epsdou max(U a, epsdou b){return a>b.v ? a: b.v;}
// template<class U> epsdou min(epsdou a, U b){return a.v<b ? a.v: b;}
// template<class U> epsdou min(U a, epsdou b){return a<b.v ? a: b.v;}
#undef double
#define double epsdou
#undef dou
#define dou epsdou
#endif

        template <class T = int, class A, class B = int>
        T my_pow(A a, B b = 2) {
  if (b < 0)
    return (T)1 / my_pow<T>(a, -b);
#if __cplusplus >= 201703L
  if constexpr (is_floating_point<T>::value) {
    return pow((T)a, (T)b);
  } else if constexpr (is_floating_point<A>::value) {
    assert2(
        0,
        "pow <not dou>(dou, )"); /*return
                                    0;しない方がコンパイル前に(voidを受け取るので)エラーが出ていいかも*/
  } else if constexpr (is_floating_point<B>::value) {
    assert2(
        0,
        "pow <not dou>(, dou)"); /*return
                                    0;しない方がコンパイル前に(voidを受け取るので)エラーが出ていいかも*/
  } else {
#endif
    T ret = 1;
    T bek = a;
    while (b) {
      if (b & 1)
        ret *= bek;
      bek *= bek;
      b >>= 1;
    }
    return ret;
#if __cplusplus >= 201703L
  }
#endif
}
#define pow my_pow

#define ull unsigned long long
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;

#define fi first
#define se second
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define el else
#define elf else if
#define upd update
#define sstream stringstream

#define maxq 1
#define minq -1

#define ZERO(a) memset(a, 0, sizeof(a))
#define MINUS(a) memset(a, 0xff, sizeof(a))
#define MALLOC(type, len) (type *)malloc((len) * sizeof(type))
#define lam1(ret) [&](auto &&v) { return ret; }
#define lam2(v, ret) [&](auto &&v) { return ret; }
#define lam(...) over2(__VA_ARGS__, lam2, lam1)(__VA_ARGS__)
#define lamr(right) [&](auto &&p) { return p right; }
#define unique(v) v.erase(unique(v.begin(), v.end()), v.end());

// マクロ省略系 コンテナ
using vi = vector<ll>;
using vb = vector<bool>;
using vs = vector<string>;
using vd = vector<double>;
using vc = vector<char>;
using vp = vector<P>;
using vt = vector<T>;

// #define V vector
#define vvt0(t) vector<vector<t>>
#define vvt1(t, a) vector<vector<t>> a
#define vvt2(t, a, b) vector<vector<t>> a(b)
#define vvt3(t, a, b, c) vector<vector<t>> a(b, vector<t>(c))
#define vvt4(t, a, b, c, d) vector<vector<t>> a(b, vector<t>(c, d))

#define vv(type, ...)                                                          \\
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(type, __VA_ARGS__)
#define vvi(...) vv(ll, __VA_ARGS__)
#define vvb(...) vv(bool, __VA_ARGS__)
#define vvs(...) vv(string, __VA_ARGS__)
#define vvd(...) vv(double, __VA_ARGS__)
#define vvc(...) vv(char, __VA_ARGS__)
#define vvp(...) vv(P, __VA_ARGS__)
#define vvt(...) vv(T, __VA_ARGS__)
// optional
#define vvoi(...) vv(optional<ll>, __VA_ARGS__)

template <typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template <typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
#define vni(name, ...) auto name = make_v<ll>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)
#define vn(type, name, ...) auto name = make_v<type>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll>>
#define tos to_string
using mapi = map<ll, ll>;
using mapp = map<P, ll>;
using mapd = map<dou, ll>;
using mapc = map<char, ll>;
using maps = map<str, ll>;
using seti = set<ll>;
using setp = set<P>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<ll>;
#define uset unordered_set
#define useti unordered_set<ll, xorshift>
#define mset multiset
#define mseti multiset<ll>
#define umap unordered_map
#define mmap multimap

// 初期化子を与える
template <template <class...> class MAP, class K, class V, class... A>
struct init_map : public MAP<K, V, A...> {
  V init_value;
  init_map(V init_value = 0) : MAP<K, V, A...>(), init_value(init_value) {}
  V &operator[](const K &k) {
    if (MAP<K, V, A...>::count(k) == 0) {
      return MAP<K, V, A...>::operator[](k) = init_value;
    }
    return MAP<K, V, A...>::operator[](k);
  }
};
template <class... A> using map2 = init_map<map, A...>;
template <class... A> using umap2 = init_map<umap, A...>;
template <class... A> using umap_f2 = init_map<umap_f, A...>;

// 任意のマクロサポート用 使う度に初期化する //todo これを消す
int index_, v1_, v2_, v3_;

/*@formatter:off*/
string to_string(char c) {
  string ret = "";
  ret += c;
  return ret;
}
template <class T> class pq_min_max {
  vector<T> d;
  void make_heap() {
    for (int i = d.size(); i--;) {
      if (i & 1 && d[i - 1] < d[i])
        swap(d[i - 1], d[i]);
      int k = down(i);
      up(k, i);
    }
  }
  inline int parent(int k) const { return ((k >> 1) - 1) & ~1; }
  int down(int k) {
    int n = d.size();
    if (k & 1) { /* min heap*/
      while (2 * k + 1 < n) {
        int c = 2 * k + 3;
        if (n <= c || d[c - 2] < d[c])
          c -= 2;
        if (c < n && d[c] < d[k]) {
          swap(d[k], d[c]);
          k = c;
        } else
          break;
      }
    } else { /* max heap*/
      while (2 * k + 2 < n) {
        int c = 2 * k + 4;
        if (n <= c || d[c] < d[c - 2])
          c -= 2;
        if (c < n && d[k] < d[c]) {
          swap(d[k], d[c]);
          k = c;
        } else
          break;
      }
    }
    return k;
  }
  int up(int k, int root = 1) {
    if ((k | 1) < (int)d.size() && d[k & ~1] < d[k | 1]) {
      swap(d[k & ~1], d[k | 1]);
      k ^= 1;
    }
    int p;
    while (root < k && d[p = parent(k)] < d[k]) { /*max heap*/
      swap(d[p], d[k]);
      k = p;
    }
    while (root < k && d[k] < d[p = parent(k) | 1]) { /* min heap*/
      swap(d[p], d[k]);
      k = p;
    }
    return k;
  }

public:
  pq_min_max() {}
  pq_min_max(const vector<T> &d_) : d(d_) { make_heap(); }
  template <class Iter> pq_min_max(Iter first, Iter last) : d(first, last) {
    make_heap();
  }
  void operator+=(const T &x) {
    int k = d.size();
    d.push_back(x);
    up(k);
  }
  void pop_min() {
    if (d.size() < 3u) {
      d.pop_back();
    } else {
      swap(d[1], d.back());
      d.pop_back();
      int k = down(1);
      up(k);
    }
  }
  void pop_max() {
    if (d.size() < 2u) {
      d.pop_back();
    } else {
      swap(d[0], d.back());
      d.pop_back();
      int k = down(0);
      up(k);
    }
  }
  const T &get_min() const { return d.size() < 2u ? d[0] : d[1]; }
  const T &get_max() const { return d[0]; }
  int size() const { return d.size(); }
  bool empty() const { return d.empty(); }
};
// 小さいほうからM個取得するpq
template <class T> struct helper_pq_size {
  pq_min_max<T> q;
  T su = 0;
  int max_size = 0;
  helper_pq_size() {}
  helper_pq_size(int max_size) : max_size(max_size) {}
  void clear() {
    q = pq_min_max<T>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q += (v);
    if (sz(q) > max_size) {
      su -= q.get_max();
      q.pop_max();
    }
  }
  T sum() { return su; }
  T top() { return q.get_min(); }
  void pop() {
    su -= q.get_min();
    q.pop_min();
  }
  T poll() {
    T ret = q.get_min();
    su -= ret;
    q.pop_min();
    return ret;
  }
  ll size() { return q.size(); }
};
// 大きいほうからM個取得するpq
template <class T> struct helper_pqg_size {
  pq_min_max<T> q;
  T su = 0;
  int max_size = 0;
  helper_pqg_size() {}
  helper_pqg_size(int max_size) : max_size(max_size) {}
  void clear() {
    q = pq_min_max<T>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q += (v);
    if (sz(q) > max_size) {
      su -= q.get_min();
      q.pop_min();
    }
  }
  T sum() { return su; }
  T top() { return q.get_max(); }
  void pop() {
    su -= q.get_max();
    q.pop_max();
  }
  T poll() {
    T ret = q.get_max();
    su -= ret;
    q.pop_min();
    return ret;
  }
  ll size() { return q.size(); }
};
;

template <class T, class Container = vector<T>,
          class Compare = std::less<typename Container::value_type>>
struct helper_pqg {
  priority_queue<T, Container, Compare> q; /*小さい順*/
  T su = 0;
  helper_pqg() {}
  void clear() {
    q = priority_queue<T, vector<T>, greater<T>>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q.push(v);
  }
  T sum() { return su; }
  T top() { return q.top(); }
  void pop() {
    su -= q.top();
    q.pop();
  }
  T poll() {
    T ret = q.top();
    su -= ret;
    q.pop();
    return ret;
  }
  ll size() { return q.size(); }
};
template <class T> using helper_pq = helper_pqg<T, vector<T>, greater<T>>;
#if __cplusplus >= 201703L

// 小さいほうからsize個残る
// Tがoptionalなら空の時nullを返す
template <class T> struct pq {
  helper_pq<T> a_q;      /*大きい順*/
  helper_pq_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pq(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pq_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  // optionalなら空の時nullを返す
  T top() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q) == 0)
          return T();
        return a_q.top();
      } else {
        if (sz(b_q) == 0)
          return T();
        return b_q.top();
      }
    } else {
      if (aquery)
        return a_q.top();
      else
        return b_q.top();
    }
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  // optionalなら空の時何もしない
  void pop() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q))
          a_q.pop();
      } else {
        if (sz(b_q))
          b_q.pop();
      }
    } else {
      if (aquery)
        a_q.pop();
      else
        b_q.pop();
    }
  } /*T*/
  T poll() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q) == 0)
          return T();
        return a_q.poll();
      } else {
        if (sz(b_q) == 0)
          return T();
        return b_q.poll();
      }
    } else {
      if (aquery)
        return a_q.poll();
      else
        return b_q.poll();
    }
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
  /*@formatter:off*/
};
template <class T> struct pqg {
  helper_pqg<T> a_q;      /*大きい順*/
  helper_pqg_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pqg(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pqg_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
#else
// 小さいほうからsize個残る
template <class T> struct pq {
  helper_pq<T> a_q;      /*大きい順*/
  helper_pq_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pq(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pq_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
// 大きいほうからsize個残る
template <class T> struct pqg {
  helper_pqg<T> a_q;      /*大きい順*/
  helper_pqg_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pqg(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pqg_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
#endif
#define pqi pq<ll>
#define pqgi pqg<ll>
template <class T> string deb_tos(pq<T> &q) {
  vector<T> res;
  auto temq = q;
  while (sz(temq))
    res.push_back(temq.top()), temq.pop();
  stringstream ss;
  ss << res;
  return ss.str();
}
template <class T> string deb_tos(pqg<T> &q) {
  vector<T> res;
  auto temq = q;
  while (sz(temq))
    res.push_back(temq.top()), temq.pop();
  stringstream ss;
  ss << res;
  return ss.str();
}

/*@formatter:off*/
// マクロ 繰り返し
// ↓@オーバーロード隔離
// todo 使わないもの非表示
#define rep1(n) for (ll rep1i = 0, rep1lim = n; rep1i < rep1lim; ++rep1i)
#define rep2(i, n) for (ll i = 0, rep2lim = n; i < rep2lim; ++i)
#define rep3(i, m, n) for (ll i = m, rep3lim = n; i < rep3lim; ++i)
#define rep4(i, m, n, ad) for (ll i = m, rep4lim = n; i < rep4lim; i += ad)
// 逆順 閉区間
#define rer2(i, n) for (ll i = n; i >= 0; i--)
#define rer3(i, m, n) for (ll i = m, rer3lim = n; i >= rer3lim; i--)
#define rer4(i, m, n, dec) for (ll i = m, rer4lim = n; i >= rer4lim; i -= dec)

#ifdef use_for

// ループを一つにまとめないとフォーマットで汚くなるため
#define nex_ind1(i) i++
#define nex_ind2(i, j, J)                                                      \\
  i = (j + 1 == J) ? i + 1 : i, j = (j + 1 == J ? 0 : j + 1)
#define nex_ind3(i, j, k, J, K)                                                \\
  i = (j + 1 == J && k + 1 == K) ? i + 1 : i,                                  \\
  j = (k + 1 == K) ? (j + 1 == J ? 0 : j + 1) : j,                             \\
  k = (k + 1 == K ? 0 : k + 1)
#define nex_ind4(i, j, k, l, J, K, L)                                          \\
  i = (j + 1 == J && k + 1 == K && l + 1 == L) ? i + 1 : i,                    \\
  j = (k + 1 == K && l + 1 == L) ? (j + 1 == J ? 0 : j + 1) : j,               \\
  k = (l + 1 == L ? (k + 1 == K ? 0 : k + 1) : k), l = l + 1 == L ? 0 : l + 1
#define nex_ind5(i, j, k, l, m, J, K, L, M)                                    \\
  i = (j + 1 == J && k + 1 == K && l + 1 == L && m + 1 == M) ? i + 1 : i,      \\
  j = (k + 1 == K && l + 1 == L && m + 1 == M) ? (j + 1 == J ? 0 : j + 1) : j, \\
  k = (l + 1 == L && m + 1 == M ? (k + 1 == K ? 0 : k + 1) : k),               \\
  l = m + 1 == M ? l + 1 == L ? 0 : l + 1 : l, m = m + 1 == M ? 0 : m + 1

#define repss2(i, I) for (int i = 0; i < I; i++)
#define repss4(i, j, I, J)                                                     \\
  for (int i = (J ? 0 : I), j = 0; i < I; nex_ind2(i, j, J))
#define repss6(i, j, k, I, J, K)                                               \\
  for (int i = (J && K ? 0 : I), j = 0, k = 0; i < I; nex_ind3(i, j, k, J, K))
#define repss8(i, j, k, l, I, J, K, L)                                         \\
  for (int i = (J && K && L ? 0 : I), j = 0, k = 0, l = 0; i < I;              \\
       nex_ind4(i, j, k, l, J, K, L))
#define repss10(i, j, k, l, m, I, J, K, L, M)                                  \\
  for (int i = (J && K && L && M ? 0 : I), j = 0, k = 0, l = 0, m = 0; i < I;  \\
       nex_ind5(i, j, k, l, m, J, K, L, M))

// i,j,k...をnまで見る
#define reps2(i, n) repss2(i, n)
#define reps3(i, j, n) repss4(i, j, n, n)
#define reps4(i, j, k, n) repss6(i, j, k, n, n, n)
#define reps5(i, j, k, l, n) repss8(i, j, k, l, n, n, n, n)
template <class T>
void nex_repv2(int &i, int &j, int &I, int &J, vector<vector<T>> &s) {
  while (1) {
    j++;
    if (j >= J) {
      j = 0;
      i++;
      if (i < I) {
        J = (int)s[i].size();
      }
    }
    if (i >= I || J)
      return;
  }
}
template <class T>
void nex_repv3(int &i, int &j, int &k, int &I, int &J, int &K,
               vector<vector<vector<T>>> &s) {
  while (1) {
    k++;
    if (k >= K) {
      k = 0;
      j++;
      if (j >= J) {
        j = 0;
        i++;
        if (i >= I)
          return;
      }
    }
    J = (int)s[i].size();
    K = (int)s[i][j].size();
    if (J && K)
      return;
  }
}

#define repv_2(i, a) repss2(i, sz(a))
// 正方形である必要はない
// 直前を持つのとどっちが早いか
#define repv_3(i, j, a)                                                        \\
  for (int repvI = (int)a.size(), repvJ = (int)a[0].size(), i = 0, j = 0;      \\
       i < repvI; nex_repv2(i, j, repvI, repvJ, a))
// 箱状になっている事が要求される つまり[i] 次元目の要素数は一定
#define repv_4(i, j, k, a)                                                     \\
  for (int repvI = (int)a.size(), repvJ = (int)a[0].size(),                    \\
           repvK = (int)a[0][0].size(), i = 0, j = 0, k = 0;                   \\
       i < repvI; nex_repv3(i, j, k, repvI, repvJ, repvK, a))
#define repv_5(i, j, k, l, a)                                                  \\
  repss8(i, j, k, l, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]))
#define repv_6(i, j, k, l, m, a)                                               \\
  repss10(i, j, k, l, m, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]),         \\
          sz(a[0][0][0][0]))
#endif

template <typename T> struct has_rbegin_rend {
private:
  template <typename U>
  static auto check(U &&obj)
      -> decltype(std::rbegin(obj), std::rend(obj), std::true_type{});
  static std::false_type check(...);

public:
  static constexpr bool value = decltype(check(std::declval<T>()))::value;
};
template <typename T>
constexpr bool has_rbegin_rend_v = has_rbegin_rend<T>::value;
template <typename Iterator> class Range {
public:
  Range(Iterator &&begin, Iterator &&end) noexcept
      : m_begin(std::forward<Iterator>(begin)),
        m_end(std::forward<Iterator>(end)) {}
  Iterator begin() const noexcept { return m_begin; }
  Iterator end() const noexcept { return m_end; }

private:
  const Iterator m_begin;
  const Iterator m_end;
};
template <typename Iterator>
static inline Range<Iterator> makeRange(Iterator &&begin,
                                        Iterator &&end) noexcept {
  return Range<Iterator>{std::forward<Iterator>(begin),
                         std::forward<Iterator>(end)};
}
template <typename T>
static inline decltype(auto)
makeReversedRange(const std::initializer_list<T> &iniList) noexcept {
  return makeRange(std::rbegin(iniList), std::rend(iniList));
}
template <typename T, typename std::enable_if_t<has_rbegin_rend_v<T>,
                                                std::nullptr_t> = nullptr>
static inline decltype(auto) makeReversedRange(T &&c) noexcept {
  return makeRange(std::rbegin(c), std::rend(c));
} /* rbegin(),
     rend()を持たないものはこっちに分岐させて，エラーメッセージを少なくする*/
template <typename T, typename std::enable_if<!has_rbegin_rend<T>::value,
                                              std::nullptr_t>::type = nullptr>
static inline void makeReversedRange(T &&) noexcept {
  static_assert(has_rbegin_rend<T>::value,
                "Specified argument doesn't have reverse iterator.");
}

// #define use_for

#define form3(k, v, st) for (auto &&[k, v] : st)
#define form4(k, v, st, r)                                                     \\
  for (auto &&[k, v] : range(st.begin(), st.lower_bound(r)))
#define form5(k, v, st, l, r)                                                  \\
  for (auto &&[k, v] : range(st.lower_bound(l), st.lower_bound(r)))
#define form(...) over5(__VA_ARGS__, form5, form4, form3)(__VA_ARGS__)

#define forrm1(st)                                                             \\
  for (auto &&forrm_it = st.rbegin(); forrm_it != st.rend(); ++forrm_it)
#define forrm3(k, v, st)                                                       \\
  for (auto &&forrm_it = st.rbegin(); forrm_it != st.rend(); ++forrm_it)

// 向こう側で
//  ++itか it = st.erase(it)とする
#define fors1(st) for (auto &&it = st.begin(); it != st.end();)
#define fors2(v, st) for (auto &&it = st.begin(); it != st.end();)
#define fors3(v, st, r)                                                        \\
  for (auto &&it = st.begin(); it != st.end() && (*it) < r;)
#define fors4(v, st, l, r)                                                     \\
  for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r;)

#ifdef use_for
#define forslr3(st, a, b)                                                      \\
  for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr4(v, st, a, b)                                                   \\
  for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr5(v, st, r, a, b)                                                \\
  for (auto &&forslr_it = st.begin();                                          \\
       forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)
#define forslr6(v, st, l, r, a, b)                                             \\
  for (auto &&forslr_it = st.lower_bound(l);                                   \\
       forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)

#define fora_f_init_2(a, A) ;
#define fora_f_init_3(fora_f_i, a, A) auto &&a = A[fora_f_i];
#define fora_f_init_4(a, b, A, B)                                              \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];
#define fora_f_init_5(fora_f_i, a, b, A, B)                                    \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];
#define fora_f_init_6(a, b, c, A, B, C)                                        \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];
#define fora_f_init_7(fora_f_i, a, b, c, A, B, C)                              \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];
#define fora_f_init_8(a, b, c, d, A, B, C, D)                                  \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];                                                      \\
  auto &&d = D[fora_f_i];
#define fora_f_init_9(fora_f_i, a, b, c, d, A, B, C, D)                        \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];                                                      \\
  auto &&d = D[fora_f_i];
#define fora_f_init(...)                                                       \\
  over9(__VA_ARGS__, fora_f_init_9, fora_f_init_8, fora_f_init_7,              \\
        fora_f_init_6, fora_f_init_5, fora_f_init_4, fora_f_init_3,            \\
        fora_f_init_2)(__VA_ARGS__)

#define forr_init_2(a, A) auto &&a = A[forr_i];
#define forr_init_3(forr_i, a, A) auto &&a = A[forr_i];
#define forr_init_4(a, b, A, B)                                                \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];
#define forr_init_5(forr_i, a, b, A, B)                                        \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];
#define forr_init_6(a, b, c, A, B, C)                                          \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];
#define forr_init_7(forr_i, a, b, c, A, B, C)                                  \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];
#define forr_init_8(a, b, c, d, A, B, C, D)                                    \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];                                                        \\
  auto &&d = D[forr_i];
#define forr_init_9(forr_i, a, b, c, d, A, B, C, D)                            \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];                                                        \\
  auto &&d = D[forr_i];
#define forr_init(...)                                                         \\
  over9(__VA_ARGS__, forr_init_9, forr_init_8, forr_init_7, forr_init_6,       \\
        forr_init_5, forr_init_4, forr_init_3, forr_init_2)(__VA_ARGS__)

#define forp_init3(k, v, S)                                                    \\
  auto &&k = S[forp_i].first;                                                  \\
  auto &&v = S[forp_i].second;
#define forp_init4(forp_i, k, v, S)                                            \\
  auto &&k = S[forp_i].first;                                                  \\
  auto &&v = S[forp_i].second;
#define forp_init(...)                                                         \\
  over4(__VA_ARGS__, forp_init4, forp_init3, forp_init2,                       \\
        forp_init1)(__VA_ARGS__)

#define forrm_init(k, v, ...)                                                  \\
  auto &&k = (*forrm_it).fi;                                                   \\
  auto &&v = (*forrm_it).se;
#define fors_init(v, ...) auto &&v = (*it);

#define forlr_init(a, A, ngl, ngr)                                             \\
  auto a = A[forlr_i];                                                         \\
  auto prev = forlr_i ? A[forlr_i - 1] : ngl;                                  \\
  auto next = forlr_i + 1 < rep2lim ? A[forlr_i + 1] : ngr;
#define forslr_init4(a, A, ngl, ngr)                                           \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init5(a, A, r, ngl, ngr)                                        \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init6(a, A, l, r, ngl, ngr)                                     \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init(...)                                                       \\
  over6(__VA_ARGS__, forslr_init6, forslr_init5, forslr_init4)(__VA_ARGS__);

// こうしないとmapがおかしくなる
#define fora_f_2(a, A) for (auto &&a : A)
#define fora_f_3(fora_f_i, a, A) rep(fora_f_i, sz(A))
#define fora_f_4(a, b, A, B) rep(fora_f_i, sz(A))
#define fora_f_5(fora_f_i, a, b, A, B) rep(fora_f_i, sz(A))
#define fora_f_6(a, b, c, A, B, C) rep(fora_f_i, sz(A))
#define fora_f_7(fora_f_i, a, b, c, A, B, C) rep(fora_f_i, sz(A))
#define fora_f_8(a, b, c, d, A, B, C, D) rep(fora_f_i, sz(A))
#define fora_f_9(fora_f_i, a, b, c, d, A, B, C, D) rep(fora_f_i, sz(A))

#define forr_2(a, A) rer(forr_i, sz(A) - 1)
#define forr_3(forr_i, a, A) rer(forr_i, sz(A) - 1)
#define forr_4(a, b, A, B) rer(forr_i, sz(A) - 1)
#define forr_5(forr_i, a, b, A, B) rer(forr_i, sz(A) - 1)
#define forr_6(a, b, c, A, B, C) rer(forr_i, sz(A) - 1)
#define forr_7(forr_i, a, b, c, A, B, C) rer(forr_i, sz(A) - 1)
#define forr_8(a, b, c, d, A, B, C, D) rer(forr_i, sz(A) - 1)
#define forr_9(forr_i, a, b, c, d, A, B, C, D) rer(forr_i, sz(A) - 1)
#endif
// ↑@オーバーロード隔離

// rep系はインデックス、for系は中身
#define rep(...) over4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define rer(...) over4(__VA_ARGS__, rer4, rer3, rer2, )(__VA_ARGS__)

// 自分込みで残りがREM以上の間ループを回す
#define rem(i, N, REM) for (int i = 0; i < N - REM + 1; i++)

// char用のrep
#define repc(i, m, n) for (char i = m, repc3lim = n; i < repc3lim; ++i)
// i,j,k...をnまで見る
#define reps(...) over5(__VA_ARGS__, reps5, reps4, reps3, reps2, )(__VA_ARGS__)
#define repss(...)                                                             \\
  over10(__VA_ARGS__, repss10, a, repss8, a, repss6, a, repss4, a,             \\
         repss2)(__VA_ARGS__)
// vectorのindexを走査する
// repv(i,j,vvi)
#define repv(...)                                                              \\
  over6(__VA_ARGS__, repv_6, repv_5, repv_4, repv_3, repv_2, )(__VA_ARGS__)
#define rerv(i, A) for (int i = sz(A) - 1; i >= 0; i--)
// repvn(dp) nは次元
#define repv1(a) repv(i, a)
#define repv2(a) repv(i, j, a)
#define repv3(a) repv(i, j, k, a)
#define repv4(a) repv(i, j, k, l, a)
#ifdef use_for
#define fora_f(...)                                                            \\
  over9(__VA_ARGS__, fora_f_9, fora_f_8, fora_f_7, fora_f_6, fora_f_5,         \\
        fora_f_4, fora_f_3, fora_f_2)(__VA_ARGS__)
#endif
#define forr(...)                                                              \\
  over9(__VA_ARGS__, forr_9, forr_8, forr_7, forr_6, forr_5, forr_4, forr_3,   \\
        forr_2)(__VA_ARGS__)
// 0~N-2まで見る

#define forar_init(v, rv, A)                                                   \\
  auto &&v = A[forar_i];                                                       \\
  auto &&rv = A[forar_i + 1];
#define forar(v, rv, A) rep(forar_i, sz(A) - 1)

#if __cplusplus >= 201703L
template <size_t M_SZ, bool indexed, class Iterator, class T, class U = T,
          class V = T, class W = T>
class ite_vec_merge : public Iterator {
  std::size_t i = 0;
  vector<T> &A;
  vector<U> &B;
  vector<V> &C;
  vector<W> &D;

public:
  ite_vec_merge(Iterator ita, vector<T> &A)
      : Iterator(ita), A(A), B(A), C(A), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B)
      : Iterator(ita), A(A), B(B), C(A), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C)
      : Iterator(ita), A(A), B(B), C(C), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C,
                vector<W> &D)
      : Iterator(ita), A(A), B(B), C(C), D(D) {}
  auto &operator++() {
    ++i;
    this->Iterator::operator++();
    return *this;
  }
  auto operator*() const noexcept {
    if constexpr (!indexed && M_SZ == 1) {
      return tuple<T &>(A[i]);
    } else if constexpr (!indexed && M_SZ == 2) {
      return tuple<T &, U &>(A[i], B[i]);
    } else if constexpr (!indexed && M_SZ == 3) {
      return tuple<T &, U &, V &>(A[i], B[i], C[i]);
    } else if constexpr (!indexed && M_SZ == 4) {
      return tuple<T &, U &, V &, W &>(A[i], B[i], C[i], D[i]);
    } else if constexpr (indexed && M_SZ == 1) {
      return tuple<int, T &>(i, A[i]);
    } else if constexpr (indexed && M_SZ == 2) {
      return tuple<int, T &, U &>(i, A[i], B[i]);
    } else if constexpr (indexed && M_SZ == 3) {
      return tuple<int, T &, U &, V &>(i, A[i], B[i], C[i]);
    } else if constexpr (indexed && M_SZ == 4) {
      return tuple<int, T &, U &, V &, W &>(i, A[i], B[i], C[i], D[i]);
    } else {
      assert(0);
      return tuple<int>(i);
    }
  }
};
template <size_t M_SZ, bool indexed, class T, class U = T, class V = T,
          class W = T>
class vec_merge {
  vector<T> &a;
  vector<U> &b;
  vector<V> &c;
  vector<W> &d;

public:
  vec_merge(vector<T> &a) : a(a), b(a), c(a), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b) : a(a), b(b), c(a), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b, vector<V> &c)
      : a(a), b(b), c(c), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b, vector<V> &c, vector<W> &d)
      : a(a), b(b), c(c), d(d) {}
  auto begin() const {
    if constexpr (M_SZ == 1) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a};
    } else if constexpr (M_SZ == 2) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b};
    } else if constexpr (M_SZ == 3) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b, c};
    } else if constexpr (M_SZ == 4) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b, c, d};
    } else {
      assert(0);
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a};
    }
  }
  auto end() const {
    if constexpr (M_SZ == 1) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a};
    } else if constexpr (M_SZ == 2) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b};
    } else if constexpr (M_SZ == 3) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b, c};
    } else if constexpr (M_SZ == 4) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b, c, d};
    } else {
      assert(0);
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a};
    }
  }
};
#endif
#define fora_2(a, A) for (auto &&a : A)

#if __cplusplus >= 201703L
#define fora_3(i, a, A)                                                        \\
  for (auto [i, a] : vec_merge<1, true, decl_t<decltype(A)>>(A))
#define fora_4(a, b, A, B)                                                     \\
  for (auto [a, b] :                                                           \\
       vec_merge<2, false, decl_t<decltype(A)>, decl_t<decltype(B)>>(A, B))
#define fora_5(i, a, b, A, B)                                                  \\
  for (auto [i, a, b] :                                                        \\
       vec_merge<2, true, decl_t<decltype(A)>, decl_t<decltype(B)>>(A, B))
#define fora_6(a, b, c, A, B, C)                                               \\
  for (auto [a, b, c] :                                                        \\
       vec_merge<3, false, decl_t<decltype(A)>, decl_t<decltype(B)>,           \\
                 decl_t<decltype(C)>>(A, B, C))
#define fora_7(i, a, b, c, A, B, C)                                            \\
  for (auto [i, a, b, c] :                                                     \\
       vec_merge<3, true, decl_t<decltype(A)>, decl_t<decltype(B)>,            \\
                 decl_t<decltype(C)>>(A, B, C))
#define fora_8(a, b, c, d, A, B, C, D)                                         \\
  for (auto [a, b, c, d] :                                                     \\
       vec_merge<4, false, decl_t<decltype(A)>, decl_t<decltype(B)>,           \\
                 decl_t<decltype(C)>, decl_t<decltype(D)>>(A, B, C, D))
#define fora_9(i, a, b, c, d, A, B, C, D)                                      \\
  for (auto [i, a, b, c, d] :                                                  \\
       vec_merge<4, true, decl_t<decltype(A)>, decl_t<decltype(B)>,            \\
                 decl_t<decltype(C)>, decl_t<decltype(D)>>(A, B, C, D))
#endif
// 構造化束縛ver
// 1e5要素で40ms程度
// 遅いときはfora_fを使う
#define fora(...)                                                              \\
  over9(__VA_ARGS__, fora_9, fora_8, fora_7, fora_6, fora_5, fora_4, fora_3,   \\
        fora_2)(__VA_ARGS__)
// #define forr(v, a) for(auto&& v : makeReversedRange(a))
// 参照を取らない
/*@formatter:off*/
#ifdef use_for
template <class U> vector<U> to1d(vector<U> &a) { return a; }
template <class U> auto to1d(vector<vector<U>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      res.push_back(a2);
  return res;
}
template <class U> vector<U> to1d(vector<vector<vector<U>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        res.push_back(a3);
  return res;
}
template <class U> vector<U> to1d(vector<vector<vector<vector<U>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          res.push_back(a4);
  return res;
}
template <class U>
vector<U> to1d(vector<vector<vector<vector<vector<U>>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          for (auto &&a5 : a4)
            res.push_back(a5);
  return res;
}
template <class U>
vector<U> to1d(vector<vector<vector<vector<vector<vector<U>>>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          for (auto &&a5 : a4)
            for (auto &&a6 : a5)
              res.push_back(a6);
  return res;
}
#define forv(a, b) for (auto a : to1d(b))
// インデックスを前後含めて走査
#define ring(i, s, len)                                                        \\
  for (int i = s, prev = (s == 0) ? len - 1 : s - 1,                           \\
           next = (s == len - 1) ? 0 : s + 1, cou = 0;                         \\
       cou < len;                                                              \\
       cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
// 値と前後を見る
#define ringv(v, d)                                                            \\
  index_ = 0;                                                                  \\
  for (auto prev = d[sz(d) - 1], next = (int)d.size() > 1 ? d[1] : d[0],       \\
            v = d[0];                                                          \\
       index_ < sz(d); index_++, prev = v, v = next,                           \\
            next = (index_ >= sz(d) - 1 ? d[0] : d[index_ + 1]))
// 左右をnext prevで見る 0の左と nの右
#define forlr(v, d, banpei_l, banpei_r) rep(forlr_i, sz(d))
#endif

#define forrm(...)                                                             \\
  over5(__VA_ARGS__, forrm5, forrm4, forrm3, forrm2, forrm1)(__VA_ARGS__)
#define fors(...) over4(__VA_ARGS__, fors4, fors3, fors2, fors1)(__VA_ARGS__)
#define forslr(...)                                                            \\
  over6(__VA_ARGS__, forslr6, forslr5, forslr4, forslr3)(__VA_ARGS__)
#define forp3(k, v, st) rep(forp_i, sz(st))
#define forp4(forp_i, k, v, st) rep(forp_i, sz(st))
#define forp(...) over4(__VA_ARGS__, forp4, forp3)(__VA_ARGS__)

// to_vec(rep(i, N))のように使い
// iが走査した値を持つvectorを返す
#define to_vec2(type, my_for)                                                  \\
  [&]() {                                                                      \\
    vector<type> ret;                                                          \\
    my_for { ret.push_back(i); }                                               \\
    return ret;                                                                \\
  }()
#define to_vec1(my_for) to_vec2(int, my_for)
#define to_vec(...) over2(__VA_ARGS__, to_vec2, to_vec1)(__VA_ARGS__)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010

const double PI = 3.1415926535897932384626433832795029L;

constexpr bool ev(ll a) { return !(a & 1); }
constexpr bool od(ll a) { return (a & 1); }
//@拡張系 こう出来るべきというもの

// 埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
template <> class hash<std::pair<signed, signed>> {
public:
  size_t operator()(const std::pair<signed, signed> &x) const {
    return hash<ll>()(((ll)x.first << 32) | x.second);
  }
};
template <> class hash<std::pair<ll, ll>> {
public
    : /*大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断*/
  size_t operator()(const std::pair<ll, ll> &x) const {
    return hash<ll>()(((ll)x.first << 32) | x.second);
  }
};
} // namespace std
// stream まとめ
/*@formatter:off*/
istream &operator>>(istream &iss, P &a) {
  iss >> a.first >> a.second;
  return iss;
}
template <typename T> istream &operator>>(istream &iss, vector<T> &vec_) {
  for (T &x : vec_)
    iss >> x;
  return iss;
}
template <class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
  os << p.fi << " " << p.se;
  return os;
}
ostream &operator<<(ostream &os, T p) {
  os << get<0>(p) << " " << get<1>(p) << " " << get<2>(p);
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << get<0>(p) << " " << get<1>(p) << " " << get<2>(p) << " " << get<3>(p);
  return os;
}
template <typename T> ostream &operator<<(ostream &os, const vector<T> &vec_) {
  for (ll i = 0; i < vec_.size(); ++i)
    os << vec_[i] << (i + 1 == vec_.size() ? "" : " ");
  return os;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<vector<T>> &vec_) {
  for (ll i = 0; i < vec_.size(); ++i) {
    for (ll j = 0; j < vec_[i].size(); ++j) {
      os << vec_[i][j] << " ";
    }
    os << endl;
  }
  return os;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const unordered_map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}

template <typename T, typename U> string deb_tos(const unordered_map<T, U> &m) {
  map<T, U> a;
  for (auto &&v : m) {
    a[v.first] = v.second;
  }
  stringstream ss;
  ss << a;
  return ss.str();
}
template <class T> ostream &operator<<(ostream &os, const set<T> &s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, const mset<T> &s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, const deque<T> &a) {
  fora(v, a) { os << v << " "; }
  return os;
}
ostream &operator<<(ostream &os, const vector<vector<char>> &vec_) {
  rep(h, sz(vec_)) {
    rep(w, sz(vec_[0])) { os << vec_[h][w]; }
    os << endl;
  }
  return os;
}
ostream &operator<<(ostream &os,
                    const vector<vector<my_optional<char>>> &vec_) {
  rep(h, sz(vec_)) {
    rep(w, sz(vec_[0])) { os << vec_[h][w]; }
    os << endl;
  }
  return os;
}
// tuple "," 区切り
#if __cplusplus >= 201703L
template <class Tuple, size_t... Indicies>
void helper_os_tuple(ostream &os, const Tuple &tup,
                     index_sequence<Indicies...>) {
  stringstream ss;
  auto f = [&](auto a) { ss << ", " << a; };
  (..., f(get<Indicies>(tup)));
  os << ss.str().substr(2);
}
template <class... T> ostream &operator<<(ostream &os, const tuple<T...> &tup) {
  helper_os_tuple(os, tup, index_sequence_for<T...>{});
  return os;
}
#endif

template <class T> struct range_now {
  int l;
  vector<T> A;
  range_now(vector<T> &&A, int l) : A(A), l(l) {}
};
/*@formatter:off*/
// template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>&
// a) {fora_f(v,a)os<<v<<endl;return os;}
template <typename W, typename H> void resize(W &vec_, const H head) {
  vec_.resize(head);
}
template <typename W, typename H, typename... T>
void resize(W &vec_, const H &head, const T... tail) {
  vec_.resize(head);
  for (auto &v : vec_)
    resize(v, tail...);
}
// #define use_for_each  //_each _all_of _any_of _none_of _find_if _rfind_if
// _contains _count_if _erase_if _entry_if

#ifdef use_for_each
// todo Atcoderの過去問がc++17に対応したら
// for_each以外はconst & (呼び出し側のラムダも)
template <typename T, typename F> bool all_of2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : v) {
      if (!all_of2(v_, f))
        return false;
    }
    return true;
  } else {
    return f(v);
  }
}
template <typename T, typename F> bool any_of2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : v) {
      if (any_of2(v_, f))
        return true;
    }
    return false;
  } else {
    return f(v);
  }
}
template <typename T, typename F> bool none_of2(const T &v, F f) {
  return all_of2(v, [&](auto a) { return !f(a); });
}

// 存在しない場合
// 1次元 Nを返す
// 多次元(-1,-1,..)を返す
template <typename T, typename F, require_t(has_value_type<T>::value)>
ll find_if2(const T &v, F f) {
  rep(i, sz(v)) {
    if (f(v[i]))
      return i;
  }
  return sz(v);
}
template <typename T, typename F>
tuple<int, int> find_if2(const vector<vector<T>> &v, F f) {
  rep(i, sz(v)) {
    rep(j, sz(v[i])) {
      if (f(v[i][j])) {
        return tuple<int, int>(i, j);
      }
    }
  }
  return tuple<int, int>(-1, -1);
}
template <typename T, typename F>
auto find_if2(const vector<vector<vector<T>>> &v, F f) {
  rep(i, sz(v)) {
    if (auto ret = find_if2(v[i], f); get<0>(ret) != -1) {
      return tuple_cat(tuple<int>(i), ret);
    }
  }
  auto bad = tuple_cat(tuple<int>(-1), find_if2(v[0], f));
  return bad;
}
template <class T, class F> auto find_if2(const range_now<T> &v, F f) {
  return find_if2(v.A, f) + v.l;
}
// 存在しない場合
// 1次元 -1を返す
// 多次元-1を返す
template <typename T, typename F, require_t(has_value_type<T>::value)>
ll rfind_if2(const T &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return i;
  }
  return -1;
}
template <typename T, typename F>
tuple<int, int> rfind_if2(const vector<vector<T>> &v, F f) {
  rer(i, sz(v) - 1) {
    rer(j, sz(v[i]) - 1) {
      if (f(v[i][j])) {
        return tuple<int, int>(i, j);
      }
    }
  }
  return tuple<int, int>(-1, -1);
}
template <typename T, typename F>
auto rfind_if2(const vector<vector<vector<T>>> &v, F f) {
  rer(i, sz(v) - 1) {
    if (auto ret = rfind_if2(v[i], f); get<0>(ret) != -1) {
      return tuple_cat(tuple<int>(i), ret);
    }
  }
  auto bad = tuple_cat(tuple<int>(-1), rfind_if2(v[0], f));
  return bad;
}

// todo まとめられそう string,vector全般
template <class T> bool contains(const string &s, const T &v) {
  return s.find(v) != string::npos;
}
template <typename T> bool contains(const vector<T> &v, const T &val) {
  return std::find(v.begin(), v.end(), val) != v.end();
}
template <typename T, typename F> bool contains_if2(const vector<T> &v, F f) {
  return find_if(v.begin(), v.end(), f) != v.end();
}
template <typename T, typename F> ll count_if2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    ll ret = 0;
    for (auto &&v_ : v) {
      ret += count_if2(v_, f);
    }
    return ret;
  } else {
    return f(v);
  }
}
template <typename T, typename F> void for_each2(T &a, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : a)
      for_each2(v_, f);
  } else {
    f(a);
  }
}

template <typename W> ll count_od(const vector<W> &a) {
  return count_if2(a, [](ll v) { return v & 1; });
}
template <typename W> ll count_ev(const vector<W> &a) {
  return count_if2(a, [](ll v) { return !(v & 1); });
}
// 削除した後のvectorを返す
template <typename T, typename F> vector<T> erase_if2(const vector<T> &v, F f) {
  vector<T> nv;
  rep(i, sz(v)) {
    if (!f(v[i])) {
      nv.push_back(v[i]);
    }
  }
  return nv;
}
template <typename T, typename F>
vector<vector<T>> erase_if2(const vector<vector<T>> &v, F f) {
  vector<vector<T>> res;
  rep(i, sz(v)) { res[i] = erase_if2(v[i], f); }
  return res;
}

template <typename T, typename F> vector<T> entry_if2(const vector<T> &v, F f) {
  vector<T> nv;
  rep(i, sz(v)) {
    if (f(v[i])) {
      nv.push_back(v[i]);
    }
  }
  return nv;
}
template <typename T, typename F>
vector<vector<T>> entry_if2(const vector<vector<T>> &v, F f) {
  vector<vector<T>> res;
  rep(i, sz(v)) { res[i] = entry_if2(v[i], f); }
  return res;
}

template <typename T, typename F> ll l_rfind_if(const vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return i;
  }
  return -1;
}
template <typename T, typename F> bool l_contains_if(const vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return true;
  }
  return false;
}
template <class A, class B, class C> auto t_all_of(A a, B b, C c) {
  return std::all_of(a, b, c);
}
template <class A, class B, class C> auto t_any_of(A a, B b, C c) {
  return std::any_of(a, b, c);
}
template <class A, class B, class C> auto t_none_of(A a, B b, C c) {
  return std::none_of(a, b, c);
}
template <class A, class B, class C> auto t_find_if(A a, B b, C c) {
  return std::find_if(a, b, c);
}
template <class A, class B, class C> auto t_count_if(A a, B b, C c) {
  return std::count_if(a, b, c);
}

#define all_of_s__2(a, right) (t_all_of(ALL(a), lamr(right)))
#define all_of_s__3(a, v, siki) (t_all_of(ALL(a), [&](auto v) { return siki; }))
#define all_of_s(...) over3(__VA_ARGS__, all_of_s__3, all_of_s__2)(__VA_ARGS__)

// all_of(A, %2);
// all_of(A, a, a%2);
#define all_of__2(a, right) all_of2(a, lamr(right))
#define all_of__3(a, v, siki) all_of2(a, [&](auto v) { return siki; })
#define all_of(...) over3(__VA_ARGS__, all_of__3, all_of__2)(__VA_ARGS__)
#define all_of_f(a, f) all_of2(a, f)

#define any_of_s__2(a, right) (t_any_of(ALL(a), lamr(right)))
#define any_of_s__3(a, v, siki) (t_any_of(ALL(a), [&](auto v) { return siki; }))
#define any_of_s(...) over3(__VA_ARGS__, any_of_s__3, any_of_s__2)(__VA_ARGS__)

#define any_of__2(a, right) any_of2(a, lamr(right))
#define any_of__3(a, v, siki) any_of2(a, [&](auto v) { return siki; })
#define any_of(...) over3(__VA_ARGS__, any_of__3, any_of__2)(__VA_ARGS__)
#define any_of_f(a, f) any_of2(a, f)

#define none_of_s__2(a, right) (t_none_of(ALL(a), lamr(right)))
#define none_of_s__3(a, v, siki)                                               \\
  (t_none_of(ALL(a), [&](auto v) { return siki; }))
#define none_of_s(...)                                                         \\
  over3(__VA_ARGS__, none_of_s__3, none_of_s__2)(__VA_ARGS__)

#define none_of__2(a, right) none_of2(a, lamr(right))
#define none_of__3(a, v, siki) none_of2(a, [&](auto v) { return siki; })
#define none_of(...) over3(__VA_ARGS__, none_of__3, none_of__2)(__VA_ARGS__)
#define none_of_f(a, f) none_of2(a, f)

#define find_if_s__2(a, right) (t_find_if(ALL(a), lamr(right)) - a.begin())
#define find_if_s__3(a, v, siki)                                               \\
  (t_find_if(ALL(a), [&](auto v) { return siki; }) - a.begin())
#define find_if_s(...)                                                         \\
  over3(__VA_ARGS__, find_if_s__3, find_if_s__2)(__VA_ARGS__)

#define find_if__2(a, right) find_if2(a, lamr(right))
#define find_if__3(a, v, siki) find_if2(a, [&](auto v) { return siki; })
#define find_if__4(a, l, v, siki)                                              \\
  (find_if2(decltype(a)(a.begin() + l, a.end()),                               \\
            [&](auto v) { return siki; }) +                                    \\
   l)
#define find_if(...)                                                           \\
  over4(__VA_ARGS__, find_if__4, find_if__3, find_if__2)(__VA_ARGS__)
#define find_if_f(a, f) find_if2(a, f)

#define rfind_if_s__2(a, right) l_rfind_if(a, lamr(right))
#define rfind_if_s__3(a, v, siki) l_rfind_if(a, [&](auto v) { return siki; })
#define rfind_if_s(...)                                                        \\
  over3(__VA_ARGS__, rfind_if_s__3, rfind_if_s__2)(__VA_ARGS__)

#define rfind_if__2(a, right) rfind_if2(a, lamr(right))
#define rfind_if__3(a, v, siki) rfind_if2(a, [&](auto v) { return siki; })
#define rfind_if(...) over3(__VA_ARGS__, rfind_if__3, rfind_if__2)(__VA_ARGS__)
#define rfind_if_f(a, f) rfind_if2(a, f)

#define contains_if_s__2(a, right) l_contains_if(a, lamr(right))
#define contains_if_s__3(a, v, siki)                                           \\
  l_contains_if(a, [&](auto v) { return siki; })
#define contains_if_s(...)                                                     \\
  over3(__VA_ARGS__, contains_if_s__3, contains_if_s__2)(__VA_ARGS__)

#define contains_if__2(a, right) contains_if2(a, lamr(right))
#define contains_if__3(a, v, siki) contains_if2(a, [&](auto v) { return siki; })
#define contains_if(...)                                                       \\
  over3(__VA_ARGS__, contains_if__3, contains_if__2)(__VA_ARGS__)
#define contains_if_f(a, f) contains_if2(a, f)

#define count_if_s__2(a, right) (t_count_if(ALL(a), lamr(right)))
#define count_if_s__3(a, v, siki)                                              \\
  (t_count_if(ALL(a), [&](auto v) { return siki; }))
#define count_if_s(...)                                                        \\
  over3(__VA_ARGS__, count_if_s__3, count_if_s__2)(__VA_ARGS__)

#define count_if__2(a, right) count_if2(a, lamr(right))
#define count_if__3(a, v, siki) count_if2(a, [&](auto v) { return siki; })
#define count_if(...) over3(__VA_ARGS__, count_if__3, count_if__2)(__VA_ARGS__)
#define count_if_f(a, f) count_if2(a, f)

// vector<vi>で、viに対して操作
#define for_each_s__2(a, right)                                                \\
  do {                                                                         \\
    fora(v, a) { v right; }                                                    \\
  } while (0)
#define for_each_s__3(a, v, shori)                                             \\
  do {                                                                         \\
    fora(v, a) { shori; }                                                      \\
  } while (0)
#define for_each_s(...)                                                        \\
  over3(__VA_ARGS__, for_each_s__3, for_each_s__2)(__VA_ARGS__)

// vector<vi>で、intに対して操作
#define for_each__2(a, right) for_each2(a, lamr(right))
#define for_each__3(a, v, shori) for_each2(a, [&](auto &v) { shori; })
#define for_each(...) over3(__VA_ARGS__, for_each__3, for_each__2)(__VA_ARGS__)
#define for_each_f(a, f) for_each2(a, f);

template <class T, class F> vector<T> help_for_eached(const vector<T> &A, F f) {
  vector<T> ret = A;
  for_each(ret, v, f(v));
  return ret;
}
#define for_eached__2(a, right) help_for_eached(a, lamr(right))
#define for_eached__3(a, v, shori) help_for_eached(a, lam(v, shori))
#define for_eached(...)                                                        \\
  over3(__VA_ARGS__, for_eached__3, for_eached__2)(__VA_ARGS__)
#define for_eached_f(a, f) for_eached2(a, f);

#define each for_each
#define eached for_eached

// #define erase_if_s__2(a, right) l_erase_if2(a,lamr(right))
// #define erase_if_s__3(a, v, siki) l_erase_if2(a,[&](auto v){return siki;})
// #define erase_if_s(...)
// over3(__VA_ARGS__,erase_if_s__3,erase_if_s__2)(__VA_ARGS__)

#define erase_if__2(a, right) erase_if2(a, lamr(right))
#define erase_if__3(a, v, siki) erase_if2(a, [&](auto v) { return siki; })
#define erase_if(...) over3(__VA_ARGS__, erase_if__3, erase_if__2)(__VA_ARGS__)
#define erase_if_f(a, f) erase_if2(a, f)

// #define entry_if_s__2(a, right) l_entry_if2(a,lamr(right))
// #define entry_if_s__3(a, v, siki) l_entry_if2(a,[&](auto v){return siki;})
// #define entry_if_s(...)
// over3(__VA_ARGS__,entry_if_s__3,entry_if_s__2)(__VA_ARGS__)

#define entry_if__2(a, right) entry_if2(a, lamr(right))
#define entry_if__3(a, v, siki) entry_if2(a, [&](auto v) { return siki; })
#define entry_if(...) over3(__VA_ARGS__, entry_if__3, entry_if__2)(__VA_ARGS__)
#define entry_if_f(a, f) entry_if2(a, f)

#endif

// 機能削除 todo
// string.replace()は長さを指定するため、間違えやすい

/*@formatter:off*/
void replace(str &a, char key, char v) { replace(ALL(a), key, v); }
template <class T, class U> void my_replace(T &s, int l, int r, const U &t) {
  s.replace(l, r - l, t);
}
#define replace my_replace
template <class T, class U, class W> void replace(vector<W> &a, T key, U v) {
  rep(i, sz(a)) if (a[i] == key) a[i] = v;
}
template <class T, class U, class W>
void replace(vector<vector<W>> &A, T key, U v) {
  rep(i, sz(A)) replace(A[i], key, v);
}
void replace(str &a, char key, str v) {
  if (v == "")
    a.erase(remove(ALL(a), key), a.end());
}
// keyと同じかどうか01で置き換える
template <class T, class U> void replace(vector<T> &a, U k) {
  rep(i, sz(a)) a[i] = a[i] == k;
}
template <class T, class U> void replace(vector<vector<T>> &a, U k) {
  rep(i, sz(a)) rep(j, sz(a[0])) a[i][j] = a[i][j] == k;
}
void replace(str &a) {
  int dec = 0;
  if ('a' <= a[0] && a[0] <= 'z')
    dec = 'a';
  if ('A' <= a[0] && a[0] <= 'Z')
    dec = 'A';
  fora(v, a) { v -= dec; }
}
void replace(string &a, char key) {
  string res;
  for (auto c : a) {
    if (c != key)
      res.push_back(c);
  }
  swap(a, res);
}
void replace(str &a, str key, str v) {
  stringstream t;
  ll kn = sz(key);
  std::string::size_type Pos(a.find(key));
  ll l = 0;
  while (Pos != std::string::npos) {
    t << a.substr(l, Pos - l);
    t << v;
    l = Pos + kn;
    Pos = a.find(key, Pos + kn);
  }
  t << a.substr(l, sz(a) - l);
  a = t.str();
}

template <class T> bool is_permutation(vector<T> &a, vector<T> &b) {
  return is_permutation(ALL(a), ALL(b));
}
template <class T> bool next_permutation(vector<T> &a) {
  return next_permutation(ALL(a));
}
vi iota(ll s, ll len) {
  vi ve(len);
  iota(ALL(ve), s);
  return ve;
}

//[iterator, iterator)等と渡す
// vectorに変換もできる
#if __cplusplus >= 201703L
template <class I, class J> struct body_range {
  I itl;
  J itr;
  body_range(I it, J end) : itl(it), itr(end) {}
  I begin() { return itl; }
  I end() { return itr; }
  // 毎回コピーする
  operator vector<typename I::value_type>() {
    return vector<typename I::value_type>(itl, itr);
  }
};
template <class I, class J, require_t(is_integral_v<I>)>
vector<int> range(I l, J r) {
  return iota(l, r - l);
}
template <class I, require_t(is_integral_v<I>)> vector<int> range(I r) {
  return iota(0, r);
}
template <class I, require_t(!is_integral_v<I>)> auto range(I itl, I itr) {
  return body_range(itl, itr);
}
#endif
/*@formatter:off*/

// tuple<vecs...> = take_vt(vt)
#if __cplusplus >= 201703L
template <class Tup_l, class Tup_r, size_t... Indicies>
auto helper_take_vector_tuple(Tup_l &res, const Tup_r &rhs,
                              index_sequence<Indicies...>) {
  (..., get<Indicies>(res).emplace_back(get<Indicies>(rhs)));
}
template <class... T> auto take_vector_tuple(const vector<tuple<T...>> &elems) {
  tuple<vector<T>...> res;
  for (auto &&elem : elems) {
    helper_take_vector_tuple(res, elem, index_sequence_for<T...>{});
  }
  return res;
}
// tie_vt(A,B,C) = vt;
template <class... T> struct tie_vector_tuple {
  tuple<vector<T> &...> v;
  tie_vector_tuple(vector<T> &...args) : v(args...) {}
  template <class Tup, size_t... Indicies>
  void helper_emplace(int i, Tup &rhs, index_sequence<Indicies...>) {
    (..., (get<Indicies>(v).emplace_back(get<Indicies>(rhs))));
  }
  template <size_t... Indicies> void clear(index_sequence<Indicies...>) {
    (..., (get<Indicies>(v).clear()));
  }
  void operator=(vector<tuple<T...>> &rhs) {
    clear(index_sequence_for<T...>{});
    rep(i, sz(rhs)) { helper_emplace(i, rhs[i], index_sequence_for<T...>{}); }
  }
};
/*@formatter:off*/
// vt = make_vt(A, B, C)
template <class H, class... T>
auto make_vector_tuple(const vector<H> &head, const vector<T> &...tails) {
  vector<tuple<H, T...>> res;
  rep(i, sz(head)) { res.emplace_back(head[i], tails[i]...); }
  return res;
}
#endif
/*@formatter:off*/
#define mvt make_vector_tuple

#define tie_vt tie_vector_tuple
#define make_vt make_vector_tuple
#define take_vt take_vector_tuple

/*@formatter:off*/
// 基本ソート@0
template <class T> void sort(vector<T> &a, int l = -1, int r = -1) {
  set_lr12(l, r, sz(a));
  fast_sort(a.begin() + l, a.begin() + r);
}
template <class T> void rsort(vector<T> &a, int l = -1, int r = -1) {
  set_lr12(l, r, sz(a));
  fast_sort(a.begin() + l, a.begin() + r, greater<T>());
};
// f(tup)の大きさでソート
template <class U, class F> void sort(vector<U> &a, F f) {
  fast_sort(ALL(a), [&](U l, U r) { return f(l) < f(r); });
};
template <class U, class F> void rsort(vector<U> &a, F f) {
  fast_sort(ALL(a), [&](U l, U r) { return f(l) > f(r); });
};

enum tuple_comparator { /*tcom_less = 1728,    tcom_greater = 1729, */ fisi = 0,
                        fisd,
                        fdsi,
                        fdsd,
                        sifi,
                        sifd,
                        sdfi,
                        sdfd,
                        fisiti,
                        fisitd,
                        fisdti,
                        fisdtd,
                        fdsiti,
                        fdsitd,
                        fdsdti,
                        fdsdtd,
                        fitisi,
                        fitisd,
                        fitdsi,
                        fitdsd,
                        fdtisi,
                        fdtisd,
                        fdtdsi,
                        fdtdsd,
                        sifiti,
                        sifitd,
                        sifdti,
                        sifdtd,
                        sdfiti,
                        sdfitd,
                        sdfdti,
                        sdfdtd,
                        sitifi,
                        sitifd,
                        sitdfi,
                        sitdfd,
                        sdtifi,
                        sdtifd,
                        sdtdfi,
                        sdfdfd,
                        tifisi,
                        tifisd,
                        tifdsi,
                        tifdsd,
                        tdfisi,
                        tdfisd,
                        tdfdsi,
                        tdfdsd,
                        tisifi,
                        tisifd,
                        tisdfi,
                        tisdfd,
                        tdsifi,
                        tdsifd,
                        tdsdfi,
                        tdsdfd };

// todo 短くする
#define set_4(i0, i1, o0, o1)                                                  \\
  [&](U l, U r) {                                                              \\
    return get<i0>(l) != get<i0>(r) ? get<i0>(l) o0 get<i0>(r)                 \\
                                    : get<i1>(l) o1 get<i1>(r);                \\
  }
#define set_6(i0, i1, i2, o0, o1, o2)                                          \\
  [&](U l, U r) {                                                              \\
    return get<i0>(l) != get<i0>(r)   ? get<i0>(l) o0 get<i0>(r)               \\
           : get<i1>(l) != get<i1>(r) ? get<i1>(l) o1 get<i1>(r)               \\
                                      : get<i2>(l) o2 get<i2>(r);              \\
  }

// functionを返すため少し遅い todo
#if __cplusplus >= 201703L
template <class... T, class U = tuple<T...>>
function<bool(U, U)> get_function2(const vector<tuple<T...>> &a,
                                   tuple_comparator Compare) {
  if constexpr (sizeof...(T) >= 2) {
    int type = Compare;
    if (type == 0)
      return set_4(0, 1, <, <);
    if (type == 1)
      return set_4(0, 1, <, >);
    if (type == 2)
      return set_4(0, 1, >, <);
    if (type == 3)
      return set_4(0, 1, >, >);
    if (type == 4)
      return set_4(1, 0, <, <);
    if (type == 5)
      return set_4(1, 0, <, >);
    if (type == 6)
      return set_4(1, 0, >, <);
    if (type == 7)
      return set_4(1, 0, >, >);
  }
  if constexpr (sizeof...(T) >= 3) {
    int type = Compare - 8;
    if (type == 0)
      return set_6(0, 1, 2, <, <, <);
    else if (type == 1)
      return set_6(0, 1, 2, <, <, >);
    else if (type == 2)
      return set_6(0, 1, 2, <, >, <);
    else if (type == 3)
      return set_6(0, 1, 2, <, >, >);
    else if (type == 4)
      return set_6(0, 1, 2, >, <, <);
    else if (type == 5)
      return set_6(0, 1, 2, >, <, >);
    else if (type == 6)
      return set_6(0, 1, 2, >, >, <);
    else if (type == 7)
      return set_6(0, 1, 2, >, >, >);
    else if (type == 8)
      return set_6(0, 2, 1, <, <, <);
    else if (type == 9)
      return set_6(0, 2, 1, <, <, >);
    else if (type == 10)
      return set_6(0, 2, 1, <, >, <);
    else if (type == 11)
      return set_6(0, 2, 1, <, >, >);
    else if (type == 12)
      return set_6(0, 2, 1, >, <, <);
    else if (type == 13)
      return set_6(0, 2, 1, >, <, >);
    else if (type == 14)
      return set_6(0, 2, 1, >, >, <);
    else if (type == 15)
      return set_6(0, 2, 1, >, >, >);
    else if (type == 16)
      return set_6(1, 0, 2, <, <, <);
    else if (type == 17)
      return set_6(1, 0, 2, <, <, >);
    else if (type == 18)
      return set_6(1, 0, 2, <, >, <);
    else if (type == 19)
      return set_6(1, 0, 2, <, >, >);
    else if (type == 20)
      return set_6(1, 0, 2, >, <, <);
    else if (type == 21)
      return set_6(1, 0, 2, >, <, >);
    else if (type == 22)
      return set_6(1, 0, 2, >, >, <);
    else if (type == 23)
      return set_6(1, 0, 2, >, >, >);
    else if (type == 24)
      return set_6(1, 2, 0, <, <, <);
    else if (type == 25)
      return set_6(1, 2, 0, <, <, >);
    else if (type == 26)
      return set_6(1, 2, 0, <, >, <);
    else if (type == 27)
      return set_6(1, 2, 0, <, >, >);
    else if (type == 28)
      return set_6(1, 2, 0, >, <, <);
    else if (type == 29)
      return set_6(1, 2, 0, >, <, >);
    else if (type == 30)
      return set_6(1, 2, 0, >, >, <);
    else if (type == 31)
      return set_6(1, 2, 0, >, >, >);
    else if (type == 32)
      return set_6(2, 0, 1, <, <, <);
    else if (type == 33)
      return set_6(2, 0, 1, <, <, >);
    else if (type == 34)
      return set_6(2, 0, 1, <, >, <);
    else if (type == 35)
      return set_6(2, 0, 1, <, >, >);
    else if (type == 36)
      return set_6(2, 0, 1, >, <, <);
    else if (type == 37)
      return set_6(2, 0, 1, >, <, >);
    else if (type == 38)
      return set_6(2, 0, 1, >, >, <);
    else if (type == 39)
      return set_6(2, 0, 1, >, >, >);
    else if (type == 40)
      return set_6(2, 1, 0, <, <, <);
    else if (type == 41)
      return set_6(2, 1, 0, <, <, >);
    else if (type == 42)
      return set_6(2, 1, 0, <, >, <);
    else if (type == 43)
      return set_6(2, 1, 0, <, >, >);
    else if (type == 44)
      return set_6(2, 1, 0, >, <, <);
    else if (type == 45)
      return set_6(2, 1, 0, >, <, >);
    else if (type == 46)
      return set_6(2, 1, 0, >, >, <);
    else if (type == 47)
      return set_6(2, 1, 0, >, >, >);
  }
  return [&](U l, U r) { return true; };
}
template <class... T>
void sort(vector<tuple<T...>> &a, tuple_comparator Compare) {
  auto f = get_function2(a, Compare);
  fast_sort(ALL(a), f);
}
#endif
/*@formatter:off*/
#undef set_4
#undef set_6

#if __cplusplus >= 201703L
// sort_tuple
template <class H1, class H2, class... T>
void sortt(vector<H1> &head1, vector<H2> &head2, vector<T> &...tails) {
  auto vec_t = make_vt(head1, head2, tails...);
  sort(vec_t);
  tie_vt(head1, head2, tails...) = vec_t;
}
template <class H1, class H2, class... T>
void rsortt(vector<H1> &head1, vector<H2> &head2, vector<T> &...tails) {
  auto vec_t = make_vt(head1, head2, tails...);
  rsort(vec_t);
  tie_vt(head1, head2, tails...) = vec_t;
}
template <class F, class... T> void helper_sortt_f(F f, vector<T> &...a) {
  auto z = make_vt(a...);
  sort(z, f);
  tie_vt(a...) = z;
}
template <class F, class... T> void helper_rsortt_f(F f, vector<T> &...a) {
  auto z = make_vt(a...);
  rsort(z, f);
  tie_vt(a...) = z;
}
// sortt(a,b,c,d, f)まで対応->
template <class A, class B, class F, require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, F f) {
  helper_sortt_f(f, a, b);
}
template <class A, class B, class F, require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, F f) {
  helper_rsortt_f(f, a, b);
}
template <class A, class B, class C, class F, require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  helper_sortt_f(f, a, b, c);
}
template <class A, class B, class C, class F, require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  helper_rsortt_f(f, a, b, c);
}
template <class A, class B, class C, class D, class F,
          require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d, F f) {
  helper_sortt_f(f, a, b, c, d);
}
template <class A, class B, class C, class D, class F,
          require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d, F f) {
  helper_rsortt_f(f, a, b, c, d);
} //  sort_tuple f }
#endif

/*@formatter:off*/
// 任意引数は取らない
template <class T> vi sorti(const vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(b));
  sortt(b, ind);
  return ind;
}
template <class T> vi rsorti(const vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(b));
  rsortt(b, ind);
  return ind;
}

#if __cplusplus >= 201703L
template <class T, class F> vi sorti(const vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  if constexpr (is_same_v<F, tuple_comparator>) {
    auto f2 = get_function2(a, f);
    auto g = [&](int i, int j) { return f2(a[i], a[j]); };
    fast_sort(ALL(ind), g);
  } else {
    auto g = [&](int i) { return f(a[i]); };
    sort(ind, g);
  }
  return ind;
}
template <class T, class F, require_t(is_function_v<F>)>
vi rsorti(const vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  auto g = [&](int i) { return f(a[i]); };
  rsort(ind, g);
  return ind;
}

// 任意引数にfを渡したい場合は (make_vt(a, b, c), f)
template <class H, class... Ts> vi sortti(vector<H> head, vector<Ts>... tails) {
  vi ind = iota(0, sz(head));
  auto vec = make_vt(head, tails..., ind);
  sort(vec);
  auto rets = take_vt(vec);
  return get<sizeof...(tails) + 1>(rets);
}
template <class H, class... Ts>
vi rsortti(vector<H> head, vector<Ts>... tails) {
  vi ind = iota(0, sz(head));
  auto vec = make_vt(head, tails..., ind);
  rsort(vec);
  auto rets = take_vt(vec);
  return get<sizeof...(tails) + 1>(rets);
}
#endif
/*@formatter:off*/
//@5
void sort(string &a) { sort(ALL(a)); }
void rsort(string &a) { sort(RALL(a)); }
void sort(int &a, int &b) {
  if (a > b)
    swap(a, b);
}
void sort(int &a, int &b, int &c) {
  sort(a, b);
  sort(a, c);
  sort(b, c);
}
void rsort(int &a, int &b) {
  if (a < b)
    swap(a, b);
}
void rsort(int &a, int &b, int &c) {
  rsort(a, b);
  rsort(a, c);
  rsort(b, c);
}
template <class... T, class U> auto sorted(U head, T... a) {
  sort(head, a...);
  return head;
}
template <class... T, class U> auto rsorted(U head, T... a) {
  rsort(head, a...);
  return head;
}

template <class T> vector<T> merge(const vector<T> &a, const vector<T> &b) {
  vector<T> res;
  std::merge(ALL(a), ALL(b), back_inserter(res));
  return res;
}

// firstの値が同じ場合secondが最小の物を残す
template <class T, class U> void unique_min(vector<pair<T, U>> &a) {
  vector<pair<T, U>> res;
  if (sz(a))
    res.push_back(a[0]);
  rep(i, 1, sz(a)) {
    if (res.back().fi == a[i].fi) {
      chmi(res.back().se, a[i].se);
    } else {
      res.push_back(a[i]);
    }
  }
  swap(a, res);
}
// firstの値が同じ場合secondが最大の物を残す
template <class T, class U> void unique_max(vector<pair<T, U>> &a) {
  vector<pair<T, U>> res;
  if (sz(a))
    res.push_back(a[0]);
  rep(i, 1, sz(a)) {
    if (res.back().fi == a[i].fi) {
      chma(res.back().se, a[i].se);
    } else {
      res.push_back(a[i]);
    }
  }
  swap(a, res);
}

template <class T> bool includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(c);
  sort(d);
  return includes(ALL(c), ALL(d));
}
template <class T> bool distinct(const vector<T> &A) {
  if ((int)(A).size() == 1)
    return true;
  if ((int)(A).size() == 2)
    return A[0] != A[1];
  if ((int)(A).size() == 3)
    return (A[0] != A[1] && A[1] != A[2] && A[0] != A[2]);
  auto B = A;
  sort(B);
  int N = (B.size());
  unique(B);
  return N == (int)(B.size());
}
template <class H, class... T> bool distinct(const H &a, const T &...b) {
  return distinct(vector<H>{a, b...});
}

/*@formatter:off*/
template <class T, class U> void inc(pair<T, U> &a, U v = 1) {
  a.first += v, a.second += v;
}
template <class T, class U> void inc(T &a, U v = 1) { a += v; }
template <class T, class U = int> void inc(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T, class U> void dec(T &a, U v = 1) { a -= v; }
template <class T, class U = int> void dec(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class U> void dec(string &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T, class U, class W>
void dec(vector<T> &a, vector<U> &b, W v = 1) {
  for (auto &u : a)
    dec(u, v);
  for (auto &u : b)
    dec(u, v);
}
template <class T, class U, class W>
void dec(vector<T> &a, vector<U> &b, vector<W> &c) {
  for (auto &u : a)
    dec(u, 1);
  for (auto &u : b)
    dec(u, 1);
  for (auto &u : c)
    dec(u, 1);
}
bool ins(ll h, ll w, ll H, ll W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool san(ll l, ll v, ll r) { return l <= v && v < r; }
template <class T> bool ins(vector<T> &a, ll i, ll j = 0) {
  return san(0, i, sz(a)) && san(0, j, sz(a));
}
#define inside ins
ll u0(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u0(vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) { v = u(v); }
  return ret;
}
// todo 名前
bool d_(int a, int b) {
  if (b == 0)
    return false;
  return (a % b) == 0;
}

// エラー
void ole() {
#ifdef _DEBUG
  cerr << "ole" << endl;
  exit(0);
#endif
  string a = "a";
  rep(i, 30) a += a;
  rep(i, 1 << 17) cout << a << endl;
  cout << "OLE 出力長制限超過" << endl;
  exit(0);
}
void re(string s = "") {
  cerr << s << endl;
  assert(0 == 1);
  exit(0);
}

void tle() {
  while (inf)
    cout << inf << endl;
}

//@汎用便利関数 入力
ll in() {
  ll ret;
  cin >> ret;
  return ret;
}
template <class T> T in() {
  T ret;
  cin >> ret;
  return ret;
}
string sin() {
  string ret;
  cin >> ret;
  return ret;
}
template <class T> void in(T &head) { cin >> head; }
template <class T, class... U> void in(T &head, U &...tail) {
  cin >> head;
  in(tail...);
}

// value_typeを持つ場合呼べる
// len回要素を追加する
template <class Iterable, class T = typename Iterable::value_type>
Iterable tin(int len) {
  Iterable ret;
  T tem;
  while (len--) {
    cin >> tem;
    ret += tem;
  }
  return ret;
}
template <class T> T tin() {
  T ret;
  cin >> ret;
  return ret;
}
template <class T> T tind(int len = 0) {
  auto ret = tin<T>(len);
  dec(ret, 1);
  return ret;
}
#define din_t2(type, a)                                                        \\
  type a;                                                                      \\
  cin >> a
#define din_t3(type, a, b)                                                     \\
  type a, b;                                                                   \\
  cin >> a >> b
#define din_t4(type, a, b, c)                                                  \\
  type a, b, c;                                                                \\
  cin >> a >> b >> c
#define din_t5(type, a, b, c, d)                                               \\
  type a, b, c, d;                                                             \\
  cin >> a >> b >> c >> d
#define din_t6(type, a, b, c, d, e)                                            \\
  type a, b, c, d, e;                                                          \\
  cin >> a >> b >> c >> d >> e
#define din_t7(type, a, b, c, d, e, f)                                         \\
  type a, b, c, d, e, f;                                                       \\
  cin >> a >> b >> c >> d >> e >> f
#define din_t(...)                                                             \\
  over7(__VA_ARGS__, din_t7, din_t6, din_t5, din_t4, din_t3,                   \\
        din_t2)(__VA_ARGS__)

#define din(...) din_t(int, __VA_ARGS__)
#define d_in
#define dsig(...) din_t(signed, __VA_ARGS__)

#define dst(...) din_t(string, __VA_ARGS__)
#define dstr dst
#define d_str dst

#define dcha(...) din_t(char, __VA_ARGS__)
#define dchar dcha

#define ddou(...) din_t(double, __VA_ARGS__)

#define din1d(a)                                                               \\
  din_t2(int, a);                                                              \\
  a--
#define din2d(a, b)                                                            \\
  din_t3(int, a, b);                                                           \\
  a--, b--
#define din3d(a, b, c)                                                         \\
  din_t4(int, a, b, c);                                                        \\
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \\
  din_t5(int, a, b, c, d);                                                     \\
  a--, b--, c--, d--
#define dind(...) over4(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

/*@formatter:off*/
#ifdef _DEBUG
template <class T> void err2(T &&head) { cerr << head; }
template <class T, class... U> void err2(T &&head, U &&...tail) {
  cerr << head << " ";
  err2(tail...);
}
template <class T, class... U> void err(T &&head, U &&...tail) {
  cerr << head << " ";
  err2(tail...);
  cerr << "" << endl;
}
template <class T> void err(T &&head) { cerr << head << endl; }
void err() { cerr << "" << endl; }
// debで出力する最大長
constexpr int DEB_LEN = 20;
constexpr int DEB_LEN_H = 12;
string deb_tos(const int &v) {
  if (abs(v) == inf || abs(v) == linf)
    return "e";
  else
    return to_string(v);
}
template <class T> string deb_tos(const T &a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
string deb_tos(const P &p) {
  stringstream ss;
  ss << "{";
  ss << p.fi;
  ss << ", ";
  ss << p.se;
  ss << "}";
  return ss.str();
}
#ifdef use_epsdou
string deb_tos(const epsdou &a) { return deb_tos(a.v); }
#endif
template <class T> string deb_tos(const optional<T> &a) {
  if (a.has_value()) {
    return deb_tos(a.value());
  } else
    return "e";
}
template <class T> string deb_tos(const vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), DEB_LEN);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) {
    ss << deb_tos(a[i]);
    if (typeid(a[i]) == typeid(P)) {
      ss << endl;
    } else {
      ss << " ";
    }
  }
  return ss.str();
}
template <class T>
string deb_tos(const vector<vector<T>> &a, vi H, vi W, int key = -1) {
  stringstream ss;
  ss << endl;
  vi lens(sz(W));
  fora(h, H) {
    rep(wi, sz(W)) {
      if (sz(a[h]) <= W[wi])
        break;
      lens[wi] = max(lens[wi], sz(deb_tos(a[h][W[wi]])) + 1);
      lens[wi] = max(lens[wi], sz(deb_tos(W[wi])) + 1);
    }
  }
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  int wi = 0;
  fora(w, W) {
    ss << std::right << std::setw(lens[wi]) << w;
    wi++;
  }
  ss << "" << endl;
  rep(i, sz(W)) rep(lens[i]) ss << "_";
  rep(i, 3) ss << "_";
  ss << "" << endl;
  fora(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    int wi = 0;
    fora(w, W) {
      if (sz(a[h]) <= w)
        break;
      ss << std::right << std::setw(lens[wi]) << deb_tos(a[h][w]);
      wi++;
    }
    ss << "" << endl;
  }
  return ss.str();
}
template <class T>
string deb_tos(const vector<vector<T>> &a, ll H = inf, ll W = inf,
               int key = -1) {
  H = (H != inf) ? H : min({H, sz(a), DEB_LEN_H});
  W = 0;
  rep(h, H) { chma(W, sz(a[h])); }
  chmi(W, DEB_LEN_H);
  vi hs, ws;
  rep(h, H) { hs.push_back(h); }
  rep(w, W) { ws.push_back(w); }
  return deb_tos(a, hs, ws, key);
}
/*@formatter:off*/
template <class T>
string deb_tos(const vector<vector<vector<T>>> &a, ll H = inf) {
  stringstream ss;
  if (H == inf)
    H = DEB_LEN_H;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << deb_tos(a[i], inf, inf, i);
  }
  return ss.str();
}
template <class T>
string deb_tos(vector<set<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  vector<vector<T>> b(sz(a));
  rep(i, sz(a)) {
    fora(v, a[i]) { b[i].push_back(v); }
  }
  return deb_tos(b, H, W, key);
}
template <class T>
string deb_tos(vector<multiset<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  vector<vector<T>> b(sz(a));
  rep(i, sz(a)) {
    fora(v, a[i]) { b[i].push_back(v); }
  }
  return deb_tos(b, H, W, key);
}
template <class T> string deb_tos(const set<T> &S) {
  vector<T> b;
  fora(s, S) b += s;
  return deb_tos(b);
}
template <class T, size_t A> string deb_tos(T (&a)[A]) {
  return deb_tos(vector<T>(begin(a), end(a)));
}
template <class T, size_t A, size_t B> string deb_tos(T (&a)[A][B]) {
  return deb_tos(vector<vector<T>>(begin(a), end(a)));
}
template <class T, size_t A, size_t B, size_t C>
string deb_tos(T (&a)[A][B][C]) {
  return deb_tos(vector<vector<vector<T>>>(begin(a), end(a)));
}
/*@formatter:off*/
template <class T> void out2(T head) {
  cout << head;
  res_mes += deb_tos(head);
}
template <class T, class... U> void out2(T head, U... tail) {
  cout << head << " ";
  res_mes += deb_tos(head) + " ";
  out2(tail...);
}
template <class T, class... U> void out(T head, U... tail) {
  cout << head << " ";
  res_mes += deb_tos(head) + " ";
  out2(tail...);
  cout << "" << endl;
  res_mes += "\\n";
}
template <class T> void out(T head) {
  cout << head << endl;
  res_mes += deb_tos(head) + "\\n";
}
void out() { cout << "" << endl; }

#else
#define err(...) ;
template <class T> void out2(T &&head) { cout << head; }
template <class T, class... U> void out2(T &&head, U &&...tail) {
  cout << head << " ";
  out2(tail...);
}
template <class T, class... U> void out(T &&head, U &&...tail) {
  cout << head << " ";
  out2(tail...);
  cout << "" << endl;
}
template <class T> void out(T &&head) { cout << head << endl; }
void out() { cout << "" << endl; }

#endif

template <class H, class... T>
void outl(const vector<H> &h, const vector<T> &...t) {
  rep(i, sz(h)) out(h[i], t[i]...);
}

// テーブルをスペースなしで出力
template <class T> void outt(vector<vector<T>> &a) {
  rep(i, sz(a)) {
    rep(j, sz(a[i])) { cout << a[i][j]; }
    cout << endl;
  }
}
// int型をbit表記で出力
void outb(int a) { cout << bitset<20>(a) << endl; }

/*@formatter:off*/
template <class T> void na(vector<T> &a, ll n) {
  a.resize(n);
  rep(i, n) cin >> a[i];
}
template <class T> void na(set<T> &a, ll n) { rep(i, n) a.insert(in()); }
#define dna(a, n)                                                              \\
  vi a;                                                                        \\
  na(a, n); /*nを複数使うと n==in()の時バグる事に注意*/
#define dnad(a, n)                                                             \\
  vi a;                                                                        \\
  nad(a, n);
template <class T> void nao(vector<T> &a, ll n) {
  a.resize(n + 1);
  a[0] = 0;
  rep(i, n) cin >> a[i + 1];
}
template <class T> void naod(vector<T> &a, ll n) {
  a.resize(n + 1);
  a[0] = 0;
  rep(i, n) cin >> a[i + 1], a[i + 1]--;
}
template <class T> void nad(vector<T> &a, ll n) {
  a.resize(n);
  rep(i, n) cin >> a[i], a[i]--;
}
template <class T> void nad(set<T> &a, ll n) { rep(i, n) a.insert(in() - 1); }
template <class T, class U> void na2(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n);
  b.resize(n);
  rep(i, n) cin >> a[i] >> b[i];
}
template <class T, class U> void na2(set<T> &a, set<U> &b, ll n) {
  rep(i, n) {
    a.insert(in());
    b.insert(in());
  }
}
#define dna2(a, b, n)                                                          \\
  vi a, b;                                                                     \\
  na2(a, b, n);
template <class T, class U> void nao2(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n + 1);
  b.resize(n + 1);
  a[0] = b[0] = 0;
  rep(i, n) cin >> a[i + 1] >> b[i + 1];
}
#define dnao2(a, b, n)                                                         \\
  vi a, b;                                                                     \\
  nao2(a, b, n);
template <class T, class U> void na2d(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n);
  b.resize(n);
  rep(i, n) cin >> a[i] >> b[i], a[i]--, b[i]--;
}
#define dna2d(a, b, n)                                                         \\
  vi a, b;                                                                     \\
  na2d(a, b, n);
template <class T, class U, class W>
void na3(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i];
}
#define dna3(a, b, c, n)                                                       \\
  vi a, b, c;                                                                  \\
  na3(a, b, c, n);
template <class T, class U, class W>
void na3d(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;
}
#define dna3d(a, b, c, n)                                                      \\
  vi a, b, c;                                                                  \\
  na3d(a, b, c, n);
template <class T, class U, class W, class X>
void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  d.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i] >> d[i];
}
#define dna4(a, b, c, d, n)                                                    \\
  vi a, b, c, d;                                                               \\
  na4(a, b, c, d, n);
#define dna4d(a, b, c, d, n)                                                   \\
  vi a, b, c, d;                                                               \\
  na4d(a, b, c, d, n);
#define nt(a, h, w)                                                            \\
  resize(a, h, w);                                                             \\
  rep(nthi, h) rep(ntwi, w) cin >> a[nthi][ntwi];
#define ntd(a, h, w)                                                           \\
  resize(a, h, w);                                                             \\
  rep(ntdhi, h) rep(ntdwi, w) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w)                                                           \\
  resize(a, h + 2, w + 2);                                                     \\
  fill(a, '#');                                                                \\
  rep(ntphi, 1, h + 1) rep(ntpwi, 1, w + 1) cin >> a[ntphi][ntpwi];
#define dnt(S, h, w)                                                           \\
  vvi(S, h, w);                                                                \\
  nt(S, h, w);
#define dntc(S, h, w)                                                          \\
  vvc(S, h, w);                                                                \\
  nt(S, h, w);
#define dnts(S, h, w)                                                          \\
  vvs(S, h, w);                                                                \\
  nt(S, h, w);

// デバッグ
#define sp << " " <<
/*@formatter:off*/
#define deb1(x) debugName(x) << " = " << deb_tos(x)
#define deb_2(x, ...) deb1(x) << ", " << deb1(__VA_ARGS__)
#define deb_3(x, ...) deb1(x) << ", " << deb_2(__VA_ARGS__)
#define deb_4(x, ...) deb1(x) << ", " << deb_3(__VA_ARGS__)
#define deb5(x, ...) deb1(x) << ", " << deb_4(__VA_ARGS__)
#define deb6(x, ...) deb1(x) << ", " << deb5(__VA_ARGS__)
// #define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)
// #define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)
// #define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)
// #define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)

/*@formatter:off*/
#ifdef _DEBUG
#define deb(...)                                                               \\
  do {                                                                         \\
    was_deb = true;                                                            \\
    cerr << over10(__VA_ARGS__, deb10, deb9, deb8, deb7, deb6, deb5, deb_4,    \\
                   deb_3, deb_2, deb1)(__VA_ARGS__)                            \\
         << endl;                                                              \\
  } while (0)

void print_n_base(int x, int base) {
  string S;
  while (x) {
    S += (char)('0' + x % base);
    x /= base;
  }
  cerr << S << endl;
}
template <class T> void print_n_base(vector<T> X, int base) {
  cerr << endl;
  for (auto &&x : X) {
    print_n_base(x, base);
  }
  cerr << endl;
}

// n進数
#define deb2(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 2);
#define deb3(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 3);
#define deb4(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 4);

#define deb_ex_deb(x, len) debugName(x) << " = " << deb_tos(x, len)
#define call_deb_ex_deb(x, len) deb_ex_deb(x, len)
// 要素が存在する行だけ出力(vvt)
#define deb_ex(v)                                                              \\
  do {                                                                         \\
    int N = sz(v);                                                             \\
    int s = N;                                                                 \\
    int t = 0;                                                                 \\
    rep(i, N) {                                                                \\
      if (sz(v[i])) {                                                          \\
        chmi(s, i);                                                            \\
        chma(t, i);                                                            \\
      }                                                                        \\
    }                                                                          \\
    auto ex_v = sub(v, s, N);                                                  \\
    str S = deb_tos(ex_v, sz(ex_v));                                           \\
    debugName(v);                                                              \\
    cerr << " = " << endl;                                                     \\
    cerr << S << endl;                                                         \\
  } while (0);
#define debi(A)                                                                \\
  {                                                                            \\
    int len = min(sz(A), 20);                                                  \\
    was_deb = true;                                                            \\
    cerr << debugName(A) << " = " << endl;                                     \\
    rep(i, len) cerr << std::right                                             \\
                     << std::setw((int)(sz(tos(A[i])) + (i ? 1 : 0)))          \\
                     << (i % 10);                                              \\
    cerr << endl;                                                              \\
    rep(i, len) cerr << std::right                                             \\
                     << std::setw((int)(sz(tos(A[i])) + (i ? 1 : 0))) << A[i]; \\
    cerr << endl;                                                              \\
  }
template <class T, class F> string deb_tos_f(vector<T> &A, F f) {
  vi I;
  rep(i, sz(A)) {
    if (f(A[i]))
      I.push_back(i);
  }
  was_deb = true;
  stringstream ss;
  fora(ii, i, I) ss << std::right
                    << std::setw(
                           (int)(max(sz(tos(A[i])), sz(tos(i))) + (ii ? 1 : 0)))
                    << i;
  ss << endl;
  fora(ii, i, I) ss << std::right
                    << std::setw(
                           (int)(max(sz(tos(A[i])), sz(tos(i))) + (ii ? 1 : 0)))
                    << A[i];
  return ss.str();
}
template <class T, class F>
string deb_tos_f(vector<vector<T>> &a, F f, int key = -1) {
  vi hs, ws_;
  int H = sz(a), W = sz(a[0]);
  vi exh(H), exw(W);
  rep(h, H) {
    rep(w, W) {
      if (f(a[h][w])) {
        exh[h] = true;
        exw[w] = true;
      }
    }
  }
  rep(h, H) if (exh[h]) hs.push_back(h);
  rep(w, W) if (exw[w]) ws_.push_back(w);
  return deb_tos(a, hs, ws_, key);
}
template <class T, class F>
string deb_tos_f(vector<vector<vector<T>>> &a, F f) {
  stringstream ss;
  int H = sz(a);
  if (sz(a) == 0)
    return ss.str();
  int cou = 0;
  rep(i, H) {
    if (any_of(a[i], v, f(v))) {
      ss << deb_tos_f(a[i], f, i);
      cou++;
    }
    if (cou > 50)
      break;
  }
  ss << "" << endl;
  return ss.str();
}
#define debf_normal(tab, f)                                                    \\
  do {                                                                         \\
    cerr << debugName(tab) << " = " << endl;                                   \\
    cerr << deb_tos_f(tab, f) << endl;                                         \\
  } while (0);
#define debf2(tab, siki_r) debf_normal(tab, lamr(siki_r))
#define debf3(tab, v, siki) debf_normal(tab, lam(siki))
// S, sikir
// S, v, siki
#define debf(...) over3(__VA_ARGS__, debf3, debf2, debf1)(__VA_ARGS__)

#else
#define deb(...) ;
#define deb2(...) ;
#define deb3(...) ;
#define deb4(...) ;
#define deb_ex(...) ;
#define debf(...) ;
#define debi(...) ;
#endif

#define debugline(x)                                                           \\
  cerr << x << " "                                                             \\
       << "(L:" << __LINE__ << ")" << '\\n'

/*@formatter:off*/
using u32 = unsigned;
using u64 = unsigned long long;
using u128 = __uint128_t;

using bint = __int128;
std::ostream &operator<<(std::ostream &dest, __int128_t value) {
  std::ostream::sentry s(dest);
  if (s) {
    __uint128_t tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = std::end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    ll len = std::end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(std::ios_base::badbit);
    }
  }
  return dest;
}
__int128 to_bint(string &s) {
  __int128 ret = 0;
  for (ll i = 0; i < (ll)s.length(); ++i)
    if ('0' <= s[i] && s[i] <= '9')
      ret = 10 * ret + s[i] - '0';
  return ret;
}
void operator>>(istream &iss, bint &v) {
  string S;
  iss >> S;
  v = 0;
  rep(i, sz(S)) {
    v *= 10;
    v += S[i] - '0';
  }
}

// 便利関数
/*@formatter:off*/
// テスト用
#define rand xor128_
unsigned long xor128_(void) {
  static unsigned long x = 123456789, y = 362436069, z = 521288629,
                       w = 88675123;
  unsigned long t;
  t = (x ^ (x << 11));
  x = y;
  y = z;
  z = w;
  return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
}
char ranc() { return (char)('a' + rand() % 26); }
ll rand(ll min, ll max) {
  assert(min <= max);
  if (min >= 0 && max >= 0) {
    return rand() % (max + 1 - min) + min;
  } else if (max < 0) {
    return -rand(-max, -min);
  } else {
    if (rand() % 2) {
      return rand(0, max);
    } else {
      return -rand(0, -min);
    }
  }
}
ll rand(ll max) { return rand(0, max); }
template <class T> T rand(vector<T> &A) { return A[rand(sz(A) - 1)]; }
// 重複することがある
template <class T> vector<T> ranv(vector<T> &A, int N) {
  vector<T> ret(N);
  rep(i, N) { ret[i] = rand(A); }
  return ret;
}
template <class T> vector<T> ranv_unique(vector<T> &A, int N) {
  vector<T> ret(N);
  umapi was;
  rep(j, N) {
    int i;
    while (1) {
      i = rand(sz(A) - 1);
      if (was.find(i) == was.end())
        break;
    }
    ret[j] = A[i];
    was[i] = 1;
  }
  return ret;
}
vi ranv(ll n, ll min, ll max) {
  vi v(n);
  rep(i, n) v[i] = rand(min, max);
  return v;
}

/*@formatter:off*/
#ifdef _DEBUG
bool timeup(int time) {
  static bool never = true;
  if (never)
    message += "may timeup, because slow";
  never = false;
  auto end_time = system_clock::now();
  auto part = duration_cast<milliseconds>(end_time - start_time);
  auto lim = milliseconds(time);
  return part >= lim;
}
#else
bool timeup(int time) {
  auto end_time = system_clock::now();
  auto part = duration_cast<milliseconds>(end_time - start_time);
  auto lim = milliseconds(time);
  return part >= lim;
}

#endif
void set_time() { past_time = system_clock::now(); }
// MS型(millisecqnds)で返る
// set_timeをしてからの時間
auto calc_time_milli() {
  auto now = system_clock::now();
  auto part = duration_cast<milliseconds>(now - past_time);
  return part;
}
auto calc_time_micro() {
  auto now = system_clock::now();
  auto part = duration_cast<microseconds>(now - past_time);
  return part;
}
auto calc_time_nano() {
  auto now = system_clock::now();
  auto part = duration_cast<nanoseconds>(now - past_time);
  return part;
}
bool calc_time(int zikan) { return calc_time_micro() >= microseconds(zikan); }
using MS = std::chrono::microseconds;
int div(microseconds a, microseconds b) { return a / b; }
int div(nanoseconds a, nanoseconds b) {
  if (b < nanoseconds(1)) {
    return a / nanoseconds(1);
  }
  int v = a / b;
  return v;
}
// set_time();
// rep(i,lim)shori

// lim*=time_nanbai();
// rep(i,lim)shoriと使う
// 全体でmilliかかっていいときにlimを何倍してもう一回できるかを返す
int time_nanbai(int milli) {
  auto dec = duration_cast<nanoseconds>(past_time - start_time);
  auto part = calc_time_nano();
  auto can_time = nanoseconds(milli * 1000 * 1000);
  can_time -= part;
  can_time -= dec;
  return div(can_time, part);
}

/*@formatter:off*/
// #define use_rand
#ifdef use_rand
str ransu(ll n) {
  str s;
  rep(i, n) s += (char)rand('A', 'Z');
  return s;
}
str ransl(ll n) {
  str s;
  rep(i, n) s += (char)rand('a', 'z');
  return s;
}
// 単調増加
vi ranvinc(ll n, ll min, ll max) {
  vi v(n);
  bool bad = 1;
  while (bad) {
    bad = 0;
    v.resize(n);
    rep(i, n) {
      if (i && min > max - v[i - 1]) {
        bad = 1;
        break;
      }
      if (i)
        v[i] = v[i - 1] + rand(min, max - v[i - 1]);
      else
        v[i] = rand(min, max);
    }
  }
  return v;
}
// 便利 汎用
#endif

void ranvlr(ll n, ll min, ll max, vi &l, vi &r) {
  l.resize(n);
  r.resize(n);
  rep(i, n) {
    l[i] = rand(min, max);
    r[i] = l[i] + rand(0, max - l[i]);
  }
}
template <class Iterable, class T = typename Iterable::value_type>
vector<pair<T, int>> run_length(const Iterable &a) {
  vector<pair<T, int>> ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}

/*@formatter:off*/
// #define use_mgr //_goldd _goldt
#ifdef use_mgr
//->[i, f(i)]
template <class T, class U, class F>
auto mgr(T ok, U ng, const F &f,
         require_arg(is_integral<T>::value &&is_integral<U>::value)) {
  auto mid = (ok + ng);
  if (ok < ng)
    while (ng - ok > 1) {
      mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  else
    while (ok - ng > 1) {
      mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  return ok;
}
//[l, r)の中で,f(i)がtrueとなる範囲を返す okはそこに含まれる
template <class F> P mgr_range(int l, int r, F f, int ok) {
  if (f(ok) == 0) {
    out("f(ok) must true");
    re();
  }
  return mp(mgr(ok, l - 1, f), mgr(ok, r, f) + 1);
}
template <class F> auto mgrd(dou ok, dou ng, F f, int kai = 100) {
  if (ok < ng)
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  else
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  return ok;
}
template <class F> dou mgrd_time(dou ok, dou ng, F f, int time = 1980) {
  bool han = true;
  if (ok < ng)
    while (1) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
      if (timeup(time)) {
        break;
      }
    }
  else
    while (1) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
      if (timeup(time)) {
        break;
      }
    }
  return ok;
}
// todo 減らす
template <class F> auto goldd_l(ll left, ll right, F calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  ll minScore = MAX<ll>();
  ll resIndex = left;
  for (ll i = left; i < right + 1; ++i) {
    ll score = calc(i);
    if (minScore > score) {
      minScore = score;
      resIndex = i;
    }
  }
  return make_tuple(resIndex, calc(resIndex));
}
template <class F> auto goldt_l(ll left, ll right, F calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  if (left > right) {
    ll l = left;
    left = right;
    right = l;
  }
  ll maxScore = MIN<ll>();
  ll resIndex = left;
  for (ll i = left; i < right + 1; ++i) {
    ll score = calc(i);
    if (maxScore < score) {
      maxScore = score;
      resIndex = i;
    }
  }
  return make_tuple(resIndex, calc(resIndex));
}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template <class F> auto goldd_d(dou left, dou right, F calc, ll loop = 200) {
  dou GRATIO = 1.6180339887498948482045868343656;
  dou lm = left + ((right - left) / (GRATIO + 1.0));
  dou rm = lm + ((right - lm) / (GRATIO + 1.0));
  dou fl = calc(lm);
  dou fr = calc(rm); /*200にすればおそらく大丈夫*/ /*余裕なら300に*/
  ll k = 141;
  loop++;
  while (--loop) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + ((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + ((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  return make_tuple(left, calc(left));
}
template <class F> auto goldt_d(dou left, dou right, F calc, ll loop = 200) {
  double GRATIO = 1.6180339887498948482045868343656;
  dou lm = left + ((right - left) / (GRATIO + 1.0));
  dou rm = lm + ((right - lm) / (GRATIO + 1.0));
  dou fl = calc(lm);
  dou fr = calc(rm);
  loop++;
  while (--loop) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + ((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + ((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  return make_tuple(left, calc(left));
}
// l ~ rを複数の区間に分割し、極致を与えるiを返す time-20 msまで探索
template <class F> auto goldd_ls(ll l, ll r, F calc, ll time = 2000) {
  auto lim = milliseconds(time - 20);
  ll mini = 0, minv = MAX<ll>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    ll haba = (r - l + k) / k; /*((r-l+1) + k-1) /k*/
    ll nl = l;
    ll nr = l + haba;
    rep(i, k) {
      ll ni = goldd_l(nl, nr, calc);
      if (chmi(minv, calc(ni)))
        mini = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(mini, calc(mini));
}
template <class F> auto goldt_ls(ll l, ll r, F calc, ll time = 2000) {
  auto lim = milliseconds(time - 20);
  ll maxi = 0, maxv = MIN<ll>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    ll haba = (r - l + k) / k; /*((r-l+1) + k-1) /k*/
    ll nl = l;
    ll nr = l + haba;
    rep(i, k) {
      ll ni = goldt_l(nl, nr, calc);
      if (chma(maxv, calc(ni)))
        maxi = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(maxi, calc(maxi));
}
template <class F>
auto goldd_d_s(dou l, dou r, F calc, ll time = 2000) { /*20ms余裕を持つ*/
  auto lim = milliseconds(time - 20);
  dou mini = 0, minv = MAX<dou>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    dou haba = (r - l) / k;
    dou nl = l;
    dou nr = l + haba;
    rep(i, k) {
      dou ni = goldd_d(nl, nr, calc);
      if (chmi(minv, calc(ni)))
        mini = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(mini, calc(mini));
}
template <class F>
auto goldt_d_s(dou l, dou r, F calc, ll time = 2000) { /*20ms余裕を残している*/
  auto lim = milliseconds(time - 20);
  dou maxi = 0, maxv = MIN<dou>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    dou haba = (r - l) / k;
    dou nl = l;
    dou nr = l + haba;
    rep(i, k) {
      dou ni = goldt_d(nl, nr, calc);
      if (chma(maxv, calc(ni)))
        maxi = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(maxi, calc(maxi));
}
#endif

// strを整数として比較
string smax(str &a, str b) {
  if (sz(a) < sz(b)) {
    return b;
  } else if (sz(a) > sz(b)) {
    return a;
  } else if (a < b)
    return b;
  else
    return a;
}
// strを整数として比較
string smin(str &a, str b) {
  if (sz(a) > sz(b)) {
    return b;
  } else if (sz(a) < sz(b)) {
    return a;
  } else if (a > b)
    return b;
  else
    return a;
}
// エラーN
// 多次元なら(-1,-1,-1)
template <typename W, typename T> ll find(vector<W> &a, int l, const T key) {
  rep(i, l, sz(a)) if (a[i] == key) return i;
  return sz(a);
}
template <typename W, typename T> ll find(vector<W> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return sz(a);
}
template <typename W, typename T> P find(vector<vector<W>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return mp(i, j);
  return mp(-1, -1);
}
// getid(find())を返す 1次元にする
template <typename W, typename T> int findi(vector<vector<W>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return i * sz(a[0]) + j;
  return -1;
}
template <typename W, typename U>
tuple<int, int, int> find(vector<vector<vector<W>>> &a, const U key) {
  rep(i, sz(a)) rep(j, sz(a[0]))
      rep(k, sz(a[0][0])) if (a[i][j][k] ==
                              key) return tuple<int, int, int>(i, j, k);
  return tuple<int, int, int>(-1, -1, -1);
}

// 無ければN
int find(string &s, const string key) {
  int klen = sz(key);
  rep(i, sz(s) - klen + 1) {
    if (s[i] != key[0])
      continue;
    if (s.substr(i, klen) == key) {
      return i;
    }
  }
  return sz(s);
}
int find(string &s, int l, const string key) {
  int klen = sz(key);
  rep(i, l, sz(s) - klen + 1) {
    if (s[i] != key[0])
      continue;
    if (s.substr(i, klen) == key) {
      return i;
    }
  }
  return sz(s);
}
int find(string &s, const char key) {
  rep(i, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return sz(s);
}
int find(string &s, int l, const char key) {
  rep(i, l, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return sz(s);
}

// N箇所について(自分含む)右のkeyの場所を返す
template <typename W, typename T> vi finds(const W &a, const T &key) {
  int n = sz(a);
  vi rpos(n, n);
  rer(i, n - 1) {
    if (i < n - 1) {
      rpos[i] = rpos[i + 1];
    }
    if (a[i] == key)
      rpos[i] = i;
  }
  return rpos;
}
template <typename W, typename T> vi rfinds(const W &a, const T &key) {
  int n = sz(a);
  vi lpos(n, -1);
  rep(i, n) {
    if (i > 0) {
      lpos[i] = lpos[i - 1];
    }
    if (a[i] == key)
      lpos[i] = i;
  }
  return lpos;
}
// todoz
#if __cplusplus >= 201703L
template <typename W, typename T, class Iterable = typename W::value_type>
ll count(const W &a, const T &k) {
  return count_if(a, == k);
}
template <typename W, class Iterable = typename W::value_type>
vi count(const W &a) {
  vi res;
  for_each(a, v, if (sz(res) <= (int)v) res.resize((int)v + 1); res[v]++;);
  return res;
}
#else
ll count(const string &S, const char &c) {
  int res = 0;
  fora(s, S) { res += s == c; }
  return res;
}
#endif
ll count(const str &a, const str &k) {
  ll ret = 0, len = k.length();
  auto pos = a.find(k);
  while (pos != string::npos)
    pos = a.find(k, pos + len), ++ret;
  return ret;
}
/*@formatter:off*/
//'a' = 'A' = 0 として集計 既に-'a'されていても動く
vi count(str &a, int l, int r) {
  vi cou(26);
  char c = 'a';
  if ('A' <= a[l] && a[l] <= 'Z')
    c = 'A';
  if ('a' <= a[l] && a[l] <= 'z')
    c = 'a';
  else
    c = 0;
  rep(i, l, r)++ cou[a[i] - c];
  return cou;
}
#define couif count_if
// algorythm
ll rev(ll a) {
  ll res = 0;
  while (a) {
    res *= 10;
    res += a % 10;
    a /= 10;
  }
  return res;
}
template <class T> auto rev(const vector<T> &a) {
  auto b = a;
  reverse(ALL(b));
  return b;
}
/* ＼反転  */ template <class U> auto rev(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a[0]), vector<U>(sz(a)));
  rep(h, sz(a)) rep(w, sz(a[0])) b[w][h] = a[h][w];
  return b;
}
/* ｜反転  */ template <class U> auto revw(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a), vector<U>(sz(a[0])));
  int W = sz(a[0]);
  rep(h, sz(a)) rep(w, sz(a[0])) { b[h][W - 1 - w] = a[h][w]; }
  return b;
}

/* ー反転  */ template <class U> auto revh(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a), vector<U>(sz(a[0])));
  int H = sz(a);
  rep(h, sz(a)) rep(w, sz(a[0])) { b[H - 1 - h][w] = a[h][w]; }
  return b;
}

/* ／反転  */ template <class U> auto revr(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a[0]), vector<U>(sz(a)));
  int H = sz(a);
  int W = sz(a[0]);
  rep(h, sz(a)) rep(w, sz(a[0])) b[w][h] = a[H - 1 - h][W - 1 - w];
  return b;
}

auto rev(const string &a) {
  string b = a;
  reverse(ALL(b));
  return b;
}
template <class T> auto rev(const T &v, int i) { return v[sz(v) - 1 - i]; }
int rev(int N, int i) { return N - 1 - i; }
constexpr ll p10[] = {1,
                      10,
                      100,
                      1000,
                      10000,
                      100000,
                      1000000,
                      10000000,
                      100000000,
                      1000000000,
                      10000000000ll,
                      100000000000ll,
                      1000000000000ll,
                      10000000000000ll,
                      100000000000000ll,
                      1000000000000000ll,
                      10000000000000000ll,
                      100000000000000000ll,
                      1000000000000000000ll};

// 0は0桁
ll keta(ll v, int if_zero_res) {
  if (!v)
    return if_zero_res;
  if (v < p10[9]) {
    if (v < p10[4]) {
      if (v < p10[2]) {
        if (v < p10[1]) {
          if (v < p10[0])
            return 0;
          else
            return 1;
        } else
          return 2;
      } else {
        if (v < p10[3])
          return 3;
        else
          return 4;
      }
    } else {
      if (v < p10[7]) {
        if (v < p10[5])
          return 5;
        else if (v < p10[6])
          return 6;
        else
          return 7;
      } else {
        if (v < p10[8])
          return 8;
        else
          return 9;
      }
    }
  } else {
    if (v < p10[13]) {
      if (v < p10[11]) {
        if (v < p10[10])
          return 10;
        else
          return 11;
      } else {
        if (v < p10[12])
          return 12;
        else
          return 13;
      }
    } else {
      if (v < p10[15]) {
        if (v < p10[14])
          return 14;
        else
          return 15;
      } else {
        if (v < p10[17]) {
          if (v < p10[16])
            return 16;
          else
            return 17;
        } else {
          if (v < p10[18])
            return 18;
          else
            return 19;
        }
      }
    }
  }
}
#if __cplusplus >= 201703L
ll getr(ll a, ll keta) { return (a / pow<ll>(10, keta)) % 10; }
#else
ll getr(ll a, ll keta) { return (a / (int)pow(10, keta)) % 10; }
#endif
// 上から何桁目か
ll getl(ll a, ll ket) {
  int sketa = keta(a, 1);
  return getr(a, sketa - 1 - ket);
}
ll dsum(ll v, ll sin = 10) {
  ll ret = 0;
  for (; v; v /= sin)
    ret += v % sin;
  return ret;
}

ll mask10(ll v) { return p10[v] - 1; }
// 変換系

template <class T, class U>
auto to_v1(vector<reference_wrapper<U>> &ret, vector<T> &A) {
  rep(i, sz(A)) ret.push_back(A[i]);
  return ret;
}
template <class T, class U>
auto to_v1(vector<reference_wrapper<U>> &ret, vector<vector<T>> &A) {
  rep(i, sz(A)) to_v1(ret, A[i]);
  return ret;
}
// 参照付きで1次元に起こす
template <class T> auto to_v1(vector<vector<T>> &A) {
  vector<reference_wrapper<decl2_t<decltype(A)>>> ret;
  rep(i, sz(A)) to_v1(ret, A[i]);
  return ret;
}

//[v] := iとなるようなvectorを返す
// 存在しない物は-1
// 空でも動く(なぜか)
template <class T> auto keys(const T &a) {
  vector<decltype((a.begin())->fi)> res;
  for (auto &&k : a)
    res.push_back(k.fi);
  return res;
}
template <class T> auto values(const T &a) {
  vector<decltype((a.begin())->se)> res;
  for (auto &&k : a)
    res.push_back(k.se);
  return res;
}
// todo 可変長で
template <class T> constexpr T min(T a, T b, T c) {
  return a >= b ? b >= c ? c : b : a >= c ? c : a;
}
template <class T> constexpr T max(T a, T b, T c) {
  return a <= b ? b <= c ? c : b : a <= c ? c : a;
}
// 1次元のコンテナ
template <class T> int mini(const vector<T> &a) {
  return min_element(ALL(a)) - a.begin();
}
template <class T> int maxi(const vector<T> &a) {
  return max_element(ALL(a)) - a.begin();
}
template <class T> T sum(const T &A) { return A; }
template <class T> T sum(const vector<T> &A, int l = -1, int r = -1) {
  T s = 0;
  set_lr12(l, r, sz(A));
  rep(i, l, r) s += sum(A[i]);
  return s;
}
// 1次元のIterableに使える
template <class T, require_t(!has_value_type<T>::value)> T min(T &a) {
  return a;
}
template <class V, require_t(has_value_type<V>::value)>
auto min(V &a, ll s = -1, ll n = -1) {
  using Ret_t = decl2_t<V>;
  auto ret = MAX<Ret_t>();
  set_lr12(s, n, sz(a));
  if (s == n) {
    return INF<Ret_t>();
  } else {
    rep(i, s, n) { ret = min(ret, min(a[i])); }
    return ret;
  }
}
template <class T, require_t(!has_value_type<T>::value)> T max(T &a) {
  return a;
}
template <class V, require_t(has_value_type<V>::value)>
auto max(V &a, ll s = -1, ll n = -1) {
  using Ret_t = decl2_t<V>;
  auto ret = MIN<Ret_t>();
  set_lr12(s, n, sz(a));
  if (s == n) {
    return -INF<Ret_t>();
  } else {
    rep(i, s, n) { ret = max(ret, max(a[i])); }
    return ret;
  }
}
template <class T> T mul(vector<T> &v, ll t = inf) {
  T ret = v[0];
  rep(i, 1, min(t, sz(v))) ret *= v[i];
  return ret;
}
// template<class T, class U, class... W>  auto sumn(vector<T> &v, U head, W...
// tail) {    auto ret = sum(v[0], tail...);    rep(i, 1, min(sz(v), head))ret
// += sum(v[i], tail...);    return ret;} indexを持つvectorを返す

/*@formatter:off*/
template <typename W, typename T> void fill(W &xx, const T vall) { xx = vall; }
template <typename W, typename T> void fill(vector<W> &vecc, const T vall) {
  for (auto &&vx : vecc)
    fill(vx, vall);
}
template <typename W, typename T> void fill(vector<W> &xx, const T v, ll len) {
  rep(i, len) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<W> &xx, const T v, int s, ll t) {
  rep(i, s, t) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<vector<W>> &xx, T v, int sh, int th, int sw, int tw) {
  rep(h, sh, th) rep(w, sw, tw) xx[h][w] = v;
}
// #define use_fill //_sum _array _max _min
#ifdef use_fill
template <typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) {
  rep(i, N) { fill(a[i], v); }
}
template <class T, class U> void fill(vector<T> &a, const vi &ind, U val) {
  fora(v, ind) { a[v] = val; }
}
template <typename A, size_t N> auto sum(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = sum(a[l]);
  if (l == r) {
    return res - res;
  }
  rep(i, l + 1, r) res += sum(a[i]);
  return res;
}
template <typename A, size_t N> auto max(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = max(a[l]);
  if (l == r) {
    return -INF<decltype(res)>();
  }
  rep(i, l + 1, r) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> auto min(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = min(a[l]);
  if (l == r) {
    return INF<decltype(res)>();
  }
  rep(i, l + 1, r) res = min(res, min(a[i]));
  return res;
}
#endif

vi inds_(vi &a) {
  int n = max(a) + 1;
  vi ret(n, -1);
  rep(i, sz(a)) {
    assert(ret[a[i]] == -1);
    ret[a[i]] = i;
  }
  return ret;
}
void clear(PQ &q) { q = PQ(); }
void clear(priority_queue<int> &q) { q = priority_queue<int>(); }
template <class T> void clear(queue<T> &q) {
  while (q.size())
    q.pop();
}
// template<class T> T *negarr(ll size) {    T *body = (T *) malloc((size * 2 +
// 1) * sizeof(T));    return body + size;} template<class T> T *negarr2(ll h,
// ll w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new
// double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (ll i =
// 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }    double
// **a = dummy1 + h;    return a;}

template <class T> struct ruiC {
  vector<T> rui;
  ruiC(vector<T> &ru) : rui(ru) {}
  /*先頭0*/
  ruiC() : rui(1, 0) {}
  T operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruic ";
      deb(l, r);
      assert(0);
    }
    return rui[r] - rui[l];
  }
  T operator()(int r = inf) { return operator()(0, min(r, sz(rui) - 1)); }
  /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
  /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
  //    T operator[](ll i) { return rui[i + 1] - rui[i]; }
  T operator[](ll i) { return rui[i]; }
  /*0から順に追加される必要がある*/
  void operator+=(T v) { rui.push_back(rui.back() + v); }
  void add(int i, T v) {
    if (sz(rui) - 1 != i)
      ole();
    operator+=(v);
  }
  T back() { return rui.back(); }
  ll size() { return rui.size(); }
  auto begin() { return rui.begin(); }
  auto end() { return rui.end(); }
};
template <class T> string deb_tos(const ruiC<T> &a) { return deb_tos(a.rui); }
template <class T> ostream &operator<<(ostream &os, ruiC<T> a) {
  fora(v, a.rui) { os << v << " "; }
  return os;
}
template <class T> vector<T> ruiv(const vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i];
  return ret;
}
template <class T> ruiC<T> ruic(const vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
template <class T> ruiC<T> ruic() { return ruiC<T>(); }
// imoは0-indexed
// ruiは1-indexed
template <class T> vector<T> imo(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}

// #define use_rui //_imo _ruic _ruiv
#ifdef use_rui
// kと同じものの数
template <class T, class U> vi imo(const vector<T> &a, U k) {
  vi equ(sz(a));
  rep(i, sz(a)) { equ[i] = a[i] == k; }
  return imo(equ);
}
template <class T> vector<T> imox(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] ^= ret[i];
  return ret;
}
// 漸化的に最小を持つ
template <class T> vector<T> imi(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chmi(ret[i + 1], ret[i]);
  return ret;
}
template <class T> vector<T> ima(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chma(ret[i + 1], ret[i]);
  return ret;
}
template <class T> vector<T> rimi(const vector<T> &v) {
  vector<T> ret = v;
  rer(i, sz(ret) - 1, 1) chmi(ret[i - 1], ret[i]);
  return ret;
}
template <class T> vector<T> rima(const vector<T> &v) {
  vector<T> ret = v;
  rer(i, sz(ret) - 1, 1) chma(ret[i - 1], ret[i]);
  return ret;
}
template <class T> struct ruimax {
  template <typename Monoid> struct SegmentTree { /*pairで処理*/
    int sz;
    vector<Monoid> seg;
    const Monoid M1 = mp(MIN<T>(), -1);
    Monoid f(Monoid a, Monoid b) { return max(a, b); }
    void build(vector<T> &a) {
      int n = sz(a);
      sz = 1;
      while (sz < n)
        sz <<= 1;
      seg.assign(2 * sz, M1);
      rep(i, n) { seg[i + sz] = mp(a[i], i); }
      for (int k = sz - 1; k > 0; k--) {
        seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);
      }
    }
    Monoid query(int a, int b) {
      Monoid L = M1, R = M1;
      for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
        if (a & 1)
          L = f(L, seg[a++]);
        if (b & 1)
          R = f(seg[--b], R);
      }
      return f(L, R);
    }
    Monoid operator[](const int &k) const { return seg[k + sz]; }
  };

private:
  vector<T> ve;
  SegmentTree<pair<T, int>> seg;
  vector<T> rv;
  vector<int> ri;
  bool build = false;

public:
  int n;
  ruimax(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    T ma = MIN<T>();
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = -INF<T>();
    ri[0] = -1;
    rep(i, n) {
      if (chma(ma, a[i])) {
        index = i;
      }
      rv[i + 1] = ma;
      ri[i + 1] = index;
    }
  }
  T operator()(int l, int r) {
    if (!(l <= r && 0 <= l && r <= n)) {
      deb(l, r, n);
      assert(0);
    }
    if (l == 0) {
      return rv[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).first;
    }
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  T getv(int l, int r) { return operator()(l, r); }
  T getv(int r = inf) { return getv(0, min(r, n)); };
  int geti(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return ri[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).second;
    }
  }
  int geti(int r = inf) { return geti(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};
template <class T> struct ruimin {
  template <typename Monoid> struct SegmentTree { /*pairで処理*/
    int sz;
    vector<Monoid> seg;
    const Monoid M1 = mp(MAX<T>(), -1);
    Monoid f(Monoid a, Monoid b) { return min(a, b); }
    void build(vector<T> &a) {
      int n = sz(a);
      sz = 1;
      while (sz < n)
        sz <<= 1;
      seg.assign(2 * sz, M1);
      rep(i, n) { seg[i + sz] = mp(a[i], i); }
      for (int k = sz - 1; k > 0; k--) {
        seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);
      }
    }
    Monoid query(int a, int b) {
      Monoid L = M1, R = M1;
      for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
        if (a & 1)
          L = f(L, seg[a++]);
        if (b & 1)
          R = f(seg[--b], R);
      }
      return f(L, R);
    }
    Monoid operator[](const int &k) const { return seg[k + sz]; }
  };

private:
  vector<T> ve;
  SegmentTree<pair<T, int>> seg;
  vector<T> rv;
  vector<int> ri;
  bool build = false;
  int n;

public:
  ruimin(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    T mi = MAX<T>();
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = INF<T>();
    ri[0] = -1;
    rep(i, n) {
      if (chmi(mi, a[i])) {
        index = i;
      }
      rv[i + 1] = mi;
      ri[i + 1] = index;
    }
  }
  T operator()(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return rv[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).first;
    }
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  T getv(int l, int r) { return operator()(l, r); }
  T getv(int r = inf) { return getv(0, min(r, n)); };
  int geti(int l, int r) {
    {
      assert(l <= r && 0 <= l && r <= n);
      if (l == 0) {
        return ri[r];
      } else {
        if (!build)
          seg.build(ve), build = true;
        return seg.query(l, r).second;
      }
    }
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return ri[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).second;
    }
  }
  int geti(int r = inf) { return geti(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
}; /*@formatter:off*/

vvi() ruib(vi &a) {
  vvi(res, 61, sz(a) + 1);
  rep(k, 61) {
    rep(i, sz(a)) { res[k][i + 1] = res[k][i] + ((a[i] >> k) & 1); }
  }
  return res;
}
vector<ruiC<int>> ruibc(vi &a) {
  vector<ruiC<int>> ret(61);
  vvi(res, 61, sz(a));
  rep(k, 61) {
    rep(i, sz(a)) { res[k][i] = (a[i] >> k) & 1; }
    ret[k] = ruic(res[k]);
  }
  return ret;
}
// kと同じものの数
template <class T, class U> vi ruiv(T &a, U k) {
  vi ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + (a[i] == k);
  return ret;
}
template <class T, class U> ruiC<ll> ruic(T &a, U k) {
  vi ret = ruiv(a, k);
  return ruiC<ll>(ret);
}
template <class T> struct ruiC2 {
  int H;
  vector<ruiC<T>> rui;
  ruiC<T> dummy; // 変なのをよばれたときはこれを返す//todo
  ruiC2(const vector<vector<T>> &ru) : rui(sz(ru)), H(sz(ru)) {
    for (int h = 0; h < H; h++) {
      if (sz(ru[h]) == 0)
        continue;
      if (sz(dummy) == 1)
        dummy = ruic(vector<T>(sz(ru[h])));
      rui[h] = ruic(ru[h]);
    }
  }
  // WについてHを返す
  vector<T> operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruic ";
      deb(l, r);
      assert(0);
    }
    vector<T> res(H);
    for (int h = 0; h < H; h++)
      res[h] = rui[h](l, r);
    return res;
  }
  // HについてWを返す
  ruiC<T> &operator[](ll h) {
#ifdef _DEBUG
    if (h >= H) {
      message += "warning ruiC h >= H";
    }
#endif
    if (h >= H || sz(rui[h]) == 1)
      return dummy;
    else
      return rui[h];
  }
  /*@formatter:off*/
  //    vector<T> operator()(int r) { return operator()(0, r); }
  /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
  /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
  //    T operator[](ll i) { return rui[i + 1] - rui[i]; }
  /*0から順に追加される必要がある*/
  //    T back() { return rui.back(); }
  //    ll size() { return rui.size(); }
  //    auto begin(){return rui.begin();}
  //    auto end(){return rui.end();}
};

template <class T, class U> ruiC<ll> ruicou(vector<T> &a, U b) {
  vi cou(sz(a));
  rep(i, sz(a)) { cou[i] = a[i] == b; }
  return ruic(cou);
}

// メモリは形式によらず(26*N)
//  rui(l,r)でvector(26文字について, l~rのcの個数)
//  rui[h] ruic()を返す
//  添え字は'a', 'A'のまま扱う
//  (予め-='a','A'されているものが渡されたらそれに従う)
template <typename Iterable, class is_Iterable = typename Iterable::value_type>
ruiC2<ll> ruicou(const Iterable &a) {
  int H = max(a) + 1;
  vvi(cou, H);
  rep(i, sz(a)) {
    if (sz(cou[a[i]]) == 0)
      cou[a[i]].resize(sz(a));
    cou[a[i]][i] = 1;
  }
  return ruiC2<ll>(cou);
}

/*@formatter:off*/
// h query
template <class T> vector<T> imoh(vector<vector<T>> &v, int w) {
  vector<T> ret(sz(v));
  rep(h, sz(ret)) { ret[h] = v[h][w]; }
  rep(i, sz(ret) - 1) { ret[i + 1] += ret[i]; }
  return ret;
}
template <class T> vector<T> ruih(vector<vector<T>> &v, int w) {
  vector<T> ret(sz(v) + 1);
  rep(h, sz(v)) { ret[h + 1] = v[h][w]; }
  rep(i, sz(v)) { ret[i + 1] += ret[i]; }
  return ret;
}
template <class T> ruiC<T> ruihc(vector<vector<T>> &a, int w) {
  vector<T> ret = ruih(a, w);
  return ruiC<T>(ret);
}

// xor
template <class T> struct ruixC {
  vector<T> rui;
  ruixC(vector<T> &ru) : rui(ru) {}
  T operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruiXc ";
      deb(l, r);
      assert(0);
    }
    return rui[r] ^ rui[l];
  }
  T operator[](ll i) { return rui[i]; }
  T back() { return rui.back(); }
  ll size() { return rui.size(); }
};
template <class T> vector<T> ruix(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] ^ a[i];
  return ret;
}
template <class T> ruixC<ll> ruixc(vector<T> &a) {
  vi ret = ruix(a);
  return ruixC<ll>(ret);
}
// 差分を返す(累積を取ると元に戻る)
// 101なら
// 1111を返す
// 元の配列で[l, r)へのxorは
//[l]と[r]へのxorになる https://atcoder.jp/contests/abc155/tasks/abc155_f
vi ruix_diff(vi &A) {
  int N = sz(A);
  assert(N);
  vi res(N + 1);
  res[0] = A[0];
  rep(i, 1, N) { res[i] = A[i - 1] ^ A[i]; }
  res[N] = A[N - 1];
  return res;
}

template <class T> vector<T> ruim(vector<T> &a) {
  vector<T> res(a.size() + 1, 1);
  rep(i, a.size()) res[i + 1] = res[i] * a[i];
  return res;
}
// 漸化的に最小を1indexで持つ
template <class T> vector<T> ruimi(vector<T> &a) {
  ll n = sz(a);
  vector<T> ret(n + 1);
  rep(i, 1, n) {
    ret[i] = a[i - 1];
    chmi(ret[i + 1], ret[i]);
  }
  return ret;
}
// template<class T> T *rrui(vector<T> &a) {
// 右から左にかけての半開区間 (-1 n-1]
template <class T> struct rruiC {
  vector<T> rui;
  int n;
  rruiC(vector<T> &a) : n(sz(a)) {
    rui.resize(n + 1);
    rer(i, n - 1) { rui[i] = rui[i + 1] + a[i]; }
  }
  /*[r l)*/
  T operator()(int r, int l) {
    r++;
    l++;
    assert(l <= r && l >= 0 && r <= n);
    return rui[l] - rui[r];
  }
  T operator()(int l) { return operator()(n - 1, l); }
  T operator[](int i) { return operator()(i); }
};
template <class T> ostream &operator<<(ostream &os, rruiC<T> a) {
  fora(v, a.rui) { os << v << " "; }
  return os;
}
template <class T> string deb_tos(rruiC<T> &a) { return deb_tos(a.rui); }

#define rrui rruic
template <class T> rruiC<T> rruic(vector<T> &a) { return rruiC<T>(a); }

// 掛け算
template <class T> struct ruimulC {
  vector<T> rv;
  int n;
  ruimulC(vector<T> &a) : rv(a), n(sz(a)) {
    rv.resize(n + 1);
    rv[0] = 1;
    rep(i, n) { rv[i + 1] = a[i] * rv[i]; }
  }
  ruimulC() : n(0) {
    rv.resize(n + 1);
    rv[0] = 1;
  }
  void operator+=(T v) {
    rv.push_back(rv.back() * v);
    n++;
  }
  T operator()(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    return rv[r] / rv[l];
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};
template <class T> ruimulC<T> ruimul(vector<T> &a) { return ruimulC<T>(a); }
template <class T> ruimulC<T> ruimul() {
  vector<T> a;
  return ruimulC<T>(a);
}
template <class T> T *rruim(vector<T> &a) {
  ll len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  res[len - 1] = 1;
  rer(i, len - 1) res[i - 1] = res[i] * a[i];
  return res;
}
template <class T, class U, class W> T lowerBound(ruiC<T> &a, U v, W banpei) {
  return lowerBound(a.rui, v, banpei);
}
template <class T, class U, class W> T upperBound(ruiC<T> &a, U v, W banpei) {
  return upperBound(a.rui, v, banpei);
}
template <class T, class U, class W> T rlowerBound(ruiC<T> &a, U v, W banpei) {
  return rlowerBound(a.rui, v, banpei);
}
template <class T, class U, class W> T rupperBound(ruiC<T> &a, U v, W banpei) {
  return rupperBound(a.rui, v, banpei);
}
#endif

constexpr bool bget(ll m, ll keta) {
#ifdef _DEBUG
  assert(keta <= 62); // オーバーフロー 1^62までしか扱えない
#endif
  return (m >> keta) & 1;
}
// bget(n)次元
//                           NならN-1まで
vector<vi> bget2(vi &a, int keta_size) {
  vvi(res, keta_size, sz(a));
  rep(k, keta_size) {
    rep(i, sz(a)) { res[k][i] = bget(a[i], k); }
  }
  return res;
}
vi bget1(vi &a, int keta) {
  vi res(sz(a));
  rep(i, sz(a)) { res[i] = bget(a[i], keta); }
  return res;
}
#if __cplusplus >= 201703L
ll bget(ll m, ll keta, ll sinsuu) {
  m /= pow<ll>(sinsuu, keta);
  return m % sinsuu;
}
#else
ll bget(ll m, ll keta, ll sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
#endif
constexpr ll bit(ll n) {
#ifdef _DEBUG
  assert(n <= 62); // オーバーフロー 1^62までしか扱えない
#endif
  return (1LL << (n));
}
#if __cplusplus >= 201703L
ll bit(ll n, ll sinsuu) { return pow<ll>(sinsuu, n); }
#else
ll bit(ll n, ll sinsuu) { return (ll)pow(sinsuu, n); }
#endif
ll mask(ll n) { return (1ll << n) - 1; }
// aをbitに置きなおす
//{0, 2} -> 101
ll bit(const vi &a) {
  int m = 0;
  for (auto &&v : a)
    m |= bit(v);
  return m;
}

//{1, 1, 0} -> 011
// bitsetに置き換える感覚 i が立っていたら i bit目を立てる
ll bit_bool(vi &a) {
  int m = 0;
  rep(i, sz(a)) if (a[i]) m |= bit(i);
  return m;
}
#define bcou __builtin_popcountll
// 最下位ビット
ll lbit(ll n) {
  assert(n);
  return n & -n;
}
ll lbiti(ll n) {
  assert(n);
  return log2(n & -n);
}
// 最上位ビット
ll hbit(ll n) {
  assert(n);
  n |= (n >> 1);
  n |= (n >> 2);
  n |= (n >> 4);
  n |= (n >> 8);
  n |= (n >> 16);
  n |= (n >> 32);
  return n - (n >> 1);
}
ll hbiti(ll n) {
  assert(n);
  return log2(hbit(n));
}
// ll hbitk(ll n) {    ll k = 0;    rer(i, 5) {        ll a = k + (1ll << i); ll
// b = 1ll << a;        if (b <= n)k += 1ll << i;    }    return k;}
// 初期化は0を渡す
ll nextComb(ll &mask, ll n, ll r) {
  if (!mask)
    return mask = (1LL << r) - 1;
  ll x = mask & -mask; /*最下位の1*/
  ll y = mask + x;     /*連続した下の1を繰り上がらせる*/
  ll res = ((mask & ~y) / x >> 1) | y;
  if (bget(res, n))
    return mask = 0;
  else
    return mask = res;
}
// n桁以下でビットがr個立っているもののvectorを返す
vi bitCombList(ll n, ll r) {
  vi res;
  ll m = 0;
  while (nextComb(m, n, r)) {
    res.push_back(m);
  }
  return res;
}
/*over*/ #define forbit1_2(i, mas)                                             \\
  for (int forbitj = !mas ? 0 : lbit(mas), forbitm = mas,                      \\
           i = !mas ? 0 : log2(forbitj);                                       \\
       forbitm; forbitm = forbitm ^ forbitj,                                   \\
           forbitj = !forbitm ? 1 : lbit(forbitm), i = log2(forbitj))
/*over*/ #define forbit1_3(i, N, mas)                                          \\
  for (int forbitj = !mas ? 0 : lbit(mas), forbitm = mas,                      \\
           i = !mas ? 0 : log2(forbitj);                                       \\
       forbitm && i < N; forbitm = forbitm ^ forbitj,                          \\
           forbitj = !forbitm ? 1 : lbit(forbitm), i = log2(forbitj))
// masの立ってるindexを見る
//               i, [N], mas
#define forbit1(...) over3(__VA_ARGS__, forbit1_3, forbit1_2)(__VA_ARGS__)
// masが立っていないindexを見る
//               i, N, mas
#define forbit0(i, N, mas) forbit1(i, (mask(N) & (~(mas))))

// forsubをスニペットして使う
// Mの部分集合(0,M含む)を見る 3^sz(S)個ある
#define forsub_all(m, M) for (int m = M; m != -1; m = m == 0 ? -1 : (m - 1) & M)
// BASE進数
template <size_t BASE> class base_num {
  int v;

public:
  base_num(int v = 0) : v(v){};
  int operator[](int i) { return bget(v, i, BASE); }
  void operator++() { v++; }
  void operator++(signed) { v++; }
  operator int() { return v; }
};
#define base3(mas, lim, BASE) for (base_num<BASE> mas; mas < lim; mas++)
#define base2(mas, lim) base3(mas, lim, 2)
#define base(...) over3(__VA_ARGS__, base3, base2, base1)(__VA_ARGS__)

// aにある物をtrueとする
vb bool_(vi a, int n) {
  vb ret(max(max(a) + 1, n));
  rep(i, sz(a)) ret[a[i]] = true;
  return ret;
}

char itoal(ll i) { return 'a' + i; }
char itoaL(ll i) { return 'A' + i; }
ll altoi(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a';
}
ll ctoi(char c) { return c - '0'; }
char itoc(ll i) { return i + '0'; }
ll vtoi(vi &v) {
  ll res = 0;
  if (sz(v) > 18) {
    debugline("vtoi");
    deb(sz(v));
    ole();
  }
  rep(i, sz(v)) {
    res *= 10;
    res += v[i];
  }
  return res;
}
vi itov(ll i) {
  vi res;
  while (i) {
    res.push_back(i % 10);
    i /= 10;
  }
  res = rev(res);
  return res;
}
vi stov(string &a) {
  ll n = sz(a);
  vi ret(n);
  rep(i, n) { ret[i] = a[i] - '0'; }
  return ret;
}
// 基準を満たさないものは0になる
vi stov(string &a, char one) {
  ll n = sz(a);
  vi ret(n);
  rep(i, n) ret[i] = a[i] == one;
  return ret;
}
vector<vector<ll>> ctoi(vector<vector<char>> s, char c) {
  ll n = sz(s), m = sz(s[0]);
  vector<vector<ll>> res(n, vector<ll>(m));
  rep(i, n) rep(j, m) res[i][j] = s[i][j] == c;
  return res;
}
// #define use_compress
//[i] := vを返す
// aは0~n-1で置き換えられる
vi compress(vi &a) {
  vi b;
  ll len = a.size();
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
#ifdef use_compress
// ind[i] := i番目に小さい数
// map[v] := vは何番目に小さいか
vi compress(vi &a, umapi &map) {
  vi b;
  ll len = a.size();
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    ll v = a[i];
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
    map[v] = a[i];
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vi &a, vi &r) {
  vi b;
  ll len = a.size();
  fora(v, a) { b.push_back(v); }
  fora(v, r) { b.push_back(v); }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i)
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  for (ll i = 0; i < sz(r); ++i)
    r[i] = lower_bound(ALL(b), r[i]) - b.begin();
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vi &a, vi &r, vi &s) {
  vi b;
  ll len = a.size();
  fora(v, a) { b.push_back(v); }
  fora(v, r) { b.push_back(v); }
  fora(v, s) { b.push_back(v); }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i)
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  for (ll i = 0; i < sz(r); ++i)
    r[i] = lower_bound(ALL(b), r[i]) - b.begin();
  for (ll i = 0; i < sz(s); ++i)
    r[i] = lower_bound(ALL(b), s[i]) - b.begin();
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vector<vi> &a) {
  vi b;
  fora(vv, a) {
    fora(v, vv) { b.push_back(v); }
  }
  sort(b);
  unique(b);
  fora(vv, a) {
    fora(v, vv) { v = lower_bound(ALL(b), v) - b.begin(); }
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vector<vector<vi>> &a) {
  vi b;
  fora(vvv, a) {
    fora(vv, vvv) {
      fora(v, vv) { b.push_back(v); }
    }
  }
  sort(b);
  unique(b);
  fora(vvv, a) {
    fora(vv, vvv) {
      fora(v, vv) { v = lower_bound(ALL(b), v) - b.begin(); }
    }
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
void compress(ll a[], ll len) {
  vi b;
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  }
}
#endif
// 要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(ALL(a), v))
#define lowerIndex(a, v) (lower_bound(ALL(a), v) - a.begin())
#define upperIndex(a, v) (upper_bound(ALL(a), v) - a.begin())
#define rlowerIndex(a, v) (upper_bound(ALL(a), v) - a.begin() - 1)
#define rupperIndex(a, v) (lower_bound(ALL(a), v) - a.begin() - 1)

template <class T, class U, class W> T lowerBound(vector<T> &a, U v, W banpei) {
  auto it = lower_bound(a.begin(), a.end(), v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(vector<T> &a, U v, W banpei) {
  auto it = upper_bound(a.begin(), a.end(), v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W>
T rlowerBound(vector<T> &a, U v, W banpei) {
  auto it = upper_bound(a.begin(), a.end(), v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W>
T rupperBound(vector<T> &a, U v, W banpei) {
  auto it = lower_bound(a.begin(), a.end(), v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
// todo 消せないか
template <class T, class U, class W> T lowerBound(set<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(set<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T rlowerBound(set<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W> T rupperBound(set<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}

template <class T, class U, class W> T lowerBound(mset<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(mset<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T rlowerBound(mset<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W> T rupperBound(mset<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
#define next2(a) next(next(a))
#define prev2(a) prev(prev(a))

// 狭義の単調増加列 長さを返す
template <class T> int lis(vector<T> &a) {
  int n = sz(a);
  vi tail(n + 1, MAX<T>());
  rep(i, n) {
    int id = lowerIndex(tail, a[i]); /**/
    tail[id] = a[i];
  }
  return lowerIndex(tail, MAX<T>());
}
template <class T> int lis_eq(vector<T> &a) {
  int n = sz(a);
  vi tail(n + 1, MAX<T>());
  rep(i, n) {
    int id = upperIndex(tail, a[i]); /**/
    tail[id] = a[i];
  }
  return lowerIndex(tail, MAX<T>());
}

// iteratorを返す
// valueが1以上の物を返す 0は見つけ次第削除
// vを減らす場合 (*it).se--でいい
template <class T, class U, class V> auto lower_map(map<T, U> &m, V k) {
  auto ret = m.lower_bound(k);
  while (ret != m.end() && (*ret).second == 0) {
    ret = m.erase(ret);
  }
  return ret;
}
template <class T, class U, class V> auto upper_map(map<T, U> &m, V k) {
  auto ret = m.upper_bound(k);
  while (ret != m.end() && (*ret).second == 0) {
    ret = m.erase(ret);
  }
  return ret;
}
// 存在しなければエラー
template <class T, class U, class V> auto rlower_map(map<T, U> &m, V k) {
  auto ret = upper_map(m, k);
  assert(ret != m.begin());
  ret--;
  while (1) {
    if ((*ret).second != 0)
      break;
    assert(ret != m.begin());
    auto next = ret;
    --next;
    m.erase(ret);
    ret = next;
  }
  return ret;
}
template <class T, class U, class V> auto rupper_map(map<T, U> &m, V k) {
  auto ret = lower_map(m, k);
  assert(ret != m.begin());
  ret--;
  while (1) {
    if ((*ret).second != 0)
      break;
    assert(ret != m.begin());
    auto next = ret;
    --next;
    m.erase(ret);
    ret = next;
  }
  return ret;
}

template <class... T> void fin(T... s) {
  out(s...);
  exit(0);
}

// 便利 数学 math
// sub ⊂ top
bool subset(int sub, int top) { return (sub & top) == sub; }
//-180 ~ 180 degree
double atand(double h, double w) { return atan2(h, w) / PI * 180; }
//% -mの場合、最小の正の数を返す
ll mod(ll a, ll m) {
  if (m < 0)
    m *= -1;
  return (a % m + m) % m;
}
// ll pow(ll a) { return a * a; };
template <class T> T fact(int v) {
  static vector<T> fact(2, 1);
  if (sz(fact) <= v) {
    rep(i, sz(fact), v + 1) { fact.emplace_back(fact.back() * i); }
  }
  return fact[v];
}

ll comi(ll n, ll r) {
  assert(n < 100);
  static vvi(pas, 100, 100);
  if (pas[0][0])
    return pas[n][r];
  pas[0][0] = 1;
  rep(i, 1, 100) {
    pas[i][0] = 1;
    rep(j, 1, i + 1) pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];
  }
  return pas[n][r];
}
// 二項係数の偶奇を返す
int com_mod2(int n, int r) { return n == (r | (n - r)); }
double comd2(ll n, ll r) {
  static vvd(comb, 2020, 2020);
  if (comb[0][0] == 0) {
    comb[0][0] = 1;
    rep(i, 2000) {
      comb[i + 1][0] = 1;
      rep(j, 1, i + 2) { comb[i + 1][j] = comb[i][j] + comb[i][j - 1]; }
    }
  }
  return comb[n][r];
}
double comd(int n, int r) {
  if (r < 0 || r > n)
    return 0;
  if (n < 2020)
    return comd2(n, r);
  static vd fact(2, 1);
  if (sz(fact) <= n) {
    rep(i, sz(fact), n + 1) { fact.push_back(fact.back() * i); }
  }
  return fact[n] / fact[n - r] / fact[r];
}

#define gcd my_gcd
template <class T, class U, class V = typename max_type<T, U>::type>
V gcd(T a_, U b_) {
  V a = a_;
  V b = b_;
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
template <class T> T gcd(const vector<T> &b) {
  T res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
#define lcm my_lcm
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll lcm(vi a) {
  ll res = a[0];
  rep(i, 1, sz(a)) res = lcm(a[i], res);
  return res;
}
ll ceil(ll a, ll b) {
  if (b == 0) {
    debugline("ceil");
    deb(a, b);
    ole();
    return -1;
  } else if (a < 0) {
    return 0;
  } else {
    return (a + b - 1) / b;
  }
}
#define hypot my_hypot
double hypot(double dx, double dy) { return std::sqrt(dx * dx + dy * dy); }

ll sig0(int t) { return t <= 0 ? 0 : ((1 + t) * t) >> 1; }
bint sig0(bint t) { return t <= 0 ? 0 : ((1 + t) * t) >> 1; }
// ll sig(ll s, ll t) { return ((s + t) * (t - s + 1)) >> 1; }
ll sig(ll s, ll t) {
  if (s > t)
    swap(s, t);
  return sig0(t - s) + s * (t - s + 1);
}

#define tousa_i tosa_i
#define lower_tousa_i lower_tosa_i
#define upper_tousa upper_tosa
#define upper_tousa_i upper_tosa_i
ll tosa_i(ll st, ll ad, ll v) {
  assert(((v - st) % ad) == 0);
  return (v - st) / ad;
}
ll tosa_s(ll st, ll ad, ll len) { return st * len + sig0(len - 1) * ad; }

// ax + r (x は非負整数) で表せる整数のうち、v 以上となる最小の整数
ll lower_tosa(ll st, ll ad, ll v) {
  if (st >= v)
    return st;
  return (v - st + ad - 1) / ad * ad + st;
}
// 第何項か
ll lower_tosa_i(ll st, ll ad, ll v) {
  if (st >= v)
    return 0;
  return (v - st + ad - 1) / ad;
}
ll upper_tosa(ll st, ll ad, ll v) { return lower_tosa(st, ad, v + 1); }
ll upper_tosa_i(ll st, ll ad, ll v) { return lower_tosa_i(st, ad, v + 1); }

// b * res <= aを満たす [l, r)を返す div
P drange_ika(int a, int b) {
  P null_p = mp(linf, linf);
  if (b == 0) {
    if (a >= 0) {
      return mp(-linf, linf + 1) /*全て*/;
    } else {
      return null_p /*無い*/;
    }
  } else {
    if (a >= 0) {
      if (b > 0) {
        return mp(-linf, a / b + 1);
      } else {
        return mp(-(a / -b), linf + 1);
      }
    } else {
      if (b > 0) {
        return mp(-linf, -ceil(-a, b) + 1);
      } else {
        return mp(ceil(-a, -b), linf + 1);
      }
    }
  }
}

// v * v >= aとなる最小のvを返す
ll sqrt(ll a) {
  if (a < 0) {
    debugline("sqrt");
    deb(a);
    ole();
  }
  ll res = (ll)std::sqrt(a);
  while (res * res < a)
    ++res;
  return res;
}
double log(double e, double x) { return log(x) / log(e); }

/*@formatter:off*/

// 機能拡張
#define dtie(a, b)                                                             \\
  int a, b;                                                                    \\
  tie(a, b)
template <class T, class U> string to_string(T a, U b) {
  string res = "";
  res += a;
  res += b;
  return res;
}
template <class T, class U, class V> string to_string(T a, U b, V c) {
  string res = "";
  res += a;
  res += b;
  res += c;
  return res;
}
template <class T, class U, class V, class W>
string to_string(T a, U b, V c, W d) {
  string res = "";
  res += a;
  res += b;
  res += c;
  res += d;
  return res;
}
template <class T, class U, class V, class W, class X>
string to_string(T a, U b, V c, W d, X e) {
  string res = "";
  res += a;
  res += b;
  res += c;
  res += d;
  res += e;
  return res;
}
// todo stringもセットで
// find_ifはsubrを使う
string sub(string &A, int l = -1, int r = -1) {
  set_lr12(l, r, sz(A));
  assert(0 <= l && l <= r && r <= sz(A));
  return A.substr(l, r - l);
}
template <class T> vector<T> sub(const vector<T> &A, int l, int r) {
  assert(0 <= l && l <= r && r <= sz(A));
  vector<T> ret(r - l);
  std::copy(A.begin() + l, A.begin() + r, ret.begin());
  return ret;
}
template <class T> vector<T> sub(const vector<T> &A, int r) {
  return sub(A, 0, r);
}
#if __cplusplus >= 201703L
template <class I, require_t(has_value_type<I>::value)>
I subn(const I &A, int l, int len) {
  return sub(A, l, l + len);
}
#endif
template <class T, class F>
// sub2で呼ぶ
vector<T> sub(const vector<vector<T>> &A, int h, int w, int ah, int aw, F f) {
  vector<T> res;
  while (0 <= h && h < sz(A) && 0 <= w && w < sz(A[h]) && f(A[h][w])) {
    res.emplace_back(A[h][w]);
    h += ah;
    w += aw;
  }
  return res;
}
template <class T>
vector<T> sub(const vector<vector<T>> &A, int h, int w, int ah, int aw) {
  return sub(A, h, w, ah, aw, [&](T v) { return true; });
}
// range_nowを返す(find_ifでしか使われない)
#if __cplusplus >= 201703L
template <class T> auto subr(const vector<T> &A, int l) {
  return range_now(vector<T>(A.begin() + l, A.end()), l);
}
#endif
#define sub25(A, h, w, ah, aw) sub(A, h, w, ah, aw)
#define sub26(A, h, w, ah, aw, siki_r)                                         \\
  sub(A, h, w, ah, aw, [&](auto v) { return v siki_r; })
#define sub27(A, h, w, ah, aw, v, siki)                                        \\
  sub(A, h, w, ah, aw, [&](auto v) { return siki; })
#define sub2(...) over7(__VA_ARGS__, sub27, sub26, sub25)(__VA_ARGS__)

constexpr int bsetlen = 50001;
// constexpr int bsetlen = 5050;
#define bset bitset<bsetlen>
bool operator<(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] < b[i])
      return true;
    if (a[i] > b[i])
      return false;
  }
  return false;
}
bool operator>(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] > b[i])
      return true;
    if (a[i] < b[i])
      return false;
  }
  return false;
}
bool operator<=(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] < b[i])
      return true;
    if (a[i] > b[i])
      return false;
  }
  return true;
}
bool operator>=(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] > b[i])
      return true;
    if (a[i] < b[i])
      return false;
  }
  return true;
}
string operator~(string &a) {
  string res = a;
  for (auto &&c : res) {
    if (c == '0')
      c = '1';
    else if (c == '1')
      c = '0';
    else {
      cerr << "cant ~" << a << "must bit" << endl;
      exit(0);
    }
  }
  return res;
}
ostream &operator<<(ostream &os, bset &a) {
  bitset<10> b;
  vi list;
  rep(i, bsetlen) {
    if (a[i])
      list.push_back(i), b[i] = 1;
  }
  os << b << ", " << list;
  return os;
}
int hbiti(bset &a) {
  rer(i, bsetlen) {
    if (a[i])
      return i;
  }
  return -1;
}
#define hk(a, b, c) (a <= b && b < c)

// O(N/64)
bset nap(bset &a, int v) {
  bset r = a | a << v;
  return r;
}
bset nap(bset &a, bset &v) {
  bset r = a;
  rep(i, bsetlen) {
    if (v[i])
      r |= a << i;
  }
  return r;
}

template <class T> int count(set<T> &S, T l, T r) {
  assert(l < r);
  auto it = S.lower_bound(l);
  return it != S.end() && (*it) < r;
}

// template<class T> void seth(vector<vector<T>> &S, int w, vector<T> &v)
// {assert(sz(S) == sz(v));assert(w < sz(S[0]));rep(h, sz(S)) { S[h][w] = v[h];
// }}
template <class T> vector<T> geth(vector<vector<T>> &S, int w) {
  assert(w < sz(S[0]));
  vector<T> ret(sz(S));
  rep(h, sz(S)) { ret[h] = S[h][w]; }
  return ret;
}
// vector<bool>[i]は参照を返さないため、こうしないとvb[i] |=
// trueがコンパイルエラー
vb::reference operator|=(vb::reference a, bool b) { return a = a | b; }
vb::reference operator&=(vb::reference a, bool b) { return a = a & b; }
template <class T, class U>
void operator+=(pair<T, U> &a, const pair<T, U> &b) {
  a.fi += b.fi;
  a.se += b.se;
}
template <class T, class U>
pair<T, U> operator+(const pair<T, U> &a, const pair<T, U> &b) {
  return pair<T, U>(a.fi + b.fi, a.se + b.se);
}
template <class T, class U> pair<T, U> operator*(const pair<T, U> &a, int b) {
  return pair<T, U>(a.fi * b, a.se * b);
}
template <class T, class U>
pair<T, U> operator-(const pair<T, U> &a, const pair<T, U> &b) {
  return pair<T, U>(a.fi - b.fi, a.se - b.se);
}
template <class T, class U> pair<T, U> operator-(const pair<T, U> &a) {
  return pair<T, U>(-a.first, -a.second);
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rhs) {
#ifdef _DEBUG
  static bool was = false;
  if (!was)
    message += "str += 65  is  'A'  not \\"65\\" ";
  was = true;
#endif
  return lhs + (char)rhs;
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const int lhs, const basic_string<CharT, Traits, Alloc> &rhs) {
#ifdef _DEBUG
  static bool was = false;
  if (!was)
    message += "str += 65  is  'A'  not \\"65\\" ";
  was = true;
#endif
  return (char)lhs + rhs;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  lhs = lhs + rv;
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  const int rv2 = rv;
  return lhs + rv2;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  const int v = rv;
  lhs += v;
}
template <typename CharT, typename Traits, typename Alloc>
void operator*=(basic_string<CharT, Traits, Alloc> &s, int num) {
  auto bek = s;
  s = "";
  for (; num; num >>= 1) {
    if (num & 1) {
      s += bek;
    }
    bek += bek;
  }
}
template <class T, class U> void operator+=(queue<T> &a, const U &v) {
  a.push(v);
}
template <class T, class U> void operator+=(deque<T> &a, const U &v) {
  a.push_back(v);
}
template <class T>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, vector<T> &v) {
  fora(d, v) { a.push(d); }
  return a;
}
template <class T, class U>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, const U &v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, const U &v) {
  a.push(v);
  return a;
}
template <class T> set<T> &operator+=(set<T> &a, const vector<T> &v) {
  fora(d, v) { a.insert(d); }
  return a;
}
template <class T, class U> auto operator+=(set<T> &a, const U &v) {
  return a.insert(v);
}
template <class T, class U> auto operator-=(set<T> &a, const U &v) {
  return a.erase(v);
}
template <class T, class U> auto operator+=(mset<T> &a, const U &v) {
  return a.insert(v);
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, const U &v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, const U &v) {
  a.push_back(v);
  return a;
}
template <class T, class U> vector<T> operator+(U v, const vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(const vector<T> &a, const vector<T> &b) {
  vector<T> ret;
  ret = a;
  fora(v, b) { ret += v; }
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, const vector<T> &b) {
  rep(i, sz(b)) { /*こうしないとa+=aで両辺が増え続けてバグる*/
    a.push_back(b[i]);
  }
  return a;
}
template <class T, class U> map<T, U> &operator+=(map<T, U> &a, map<T, U> &b) {
  for (auto &&bv : b) {
    a[bv.first] += bv.second;
  }
  return a;
}
template <class T, class U>
vector<T> operator+(const vector<T> &a, const U &v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> auto operator+=(uset<T> &a, U v) {
  return a.insert(v);
}
template <class T> vector<T> operator%(vector<T> &a, int v) {
  vi ret(sz(a));
  rep(i, sz(a)) { ret[i] = a[i] % v; }
  return ret;
}
template <class T> vector<T> operator%=(vector<T> &a, int v) {
  rep(i, sz(a)) { a[i] %= v; }
  return a;
}
vi operator&(vi &a, vi &b) {
  assert(sz(a) == sz(b));
  vi ret(sz(a));
  rep(i, sz(a)) { ret[i] = min(a[i], b[i]); }
  return ret;
}
template <class T> void operator+=(mset<T> &a, vector<T> &v) {
  for (auto &&u : v)
    a.insert(u);
}
template <class T> void operator+=(set<T> &a, vector<T> &v) {
  for (auto &&u : v)
    a.insert(u);
}
template <class T> void operator+=(vector<T> &a, set<T> &v) {
  for (auto &&u : v)
    a.emplace_back(u);
}
template <class T> void operator+=(vector<T> &a, mset<T> &v) {
  for (auto &&u : v)
    a.emplace_back(u);
}
template <class T> vector<T> &operator-=(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-=");
    deb(a);
    deb(b);
    exit(0);
  }
  rep(i, sz(a)) a[i] -= b[i];
  return a;
}
template <class T> vector<T> operator-(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-");
    deb(a);
    deb(b);
    ole();
  }
  vector<T> res(sz(a));
  rep(i, sz(a)) res[i] = a[i] - b[i];
  return res;
}
// template<class T, class U> void operator*=(vector<T> &a, U b) {    vector<T>
// ta = a;    rep(b-1){        a+=ta;    }}
template <typename T> void erase(T &v, unsigned ll i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(T &v, unsigned ll s, unsigned ll e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <class T, class... I> auto erased(T &v, I... arg) {
  auto ret = v;
  erase(ret, arg...);
  return ret;
}
template <typename T> void pop_front(vector<T> &v) { erase(v, 0); }
template <typename T> void entry(vector<T> &v, unsigned ll s, unsigned ll e) {
  erase(v, e, sz(v));
  erase(v, 0, s);
}

template <class T, class U> void erase(map<T, U> &m, ll okl, ll ngr) {
  m.erase(m.lower_bound(okl), m.lower_bound(ngr));
}
template <class T> void erase(set<T> &m, ll okl, ll ngr) {
  m.erase(m.lower_bound(okl), m.lower_bound(ngr));
}
template <typename T> void erasen(vector<T> &v, unsigned ll s, unsigned ll n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned ll i, U t) {
  v.insert(v.begin() + i, t);
}
template <typename T, typename U> void push_front(vector<T> &v, U t) {
  v.insert(v.begin(), t);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned ll i, vector<T> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
vector<string> split(const string &a, const char deli) {
  string b = a + deli;
  ll l = 0, r = 0, n = b.size();
  vector<string> res;
  rep(i, n) {
    if (b[i] == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      l = i + 1;
    }
  }
  return res;
}
vector<string> split(const string &a, const string deli) {
  vector<string> res;
  ll kn = sz(deli);
  std::string::size_type Pos(a.find(deli));
  ll l = 0;
  while (Pos != std::string::npos) {
    if (Pos - l)
      res.push_back(a.substr(l, Pos - l));
    l = Pos + kn;
    Pos = a.find(deli, Pos + kn);
  }
  if (sz(a) - l)
    res.push_back(a.substr(l, sz(a) - l));
  return res;
}
ll stoi(string &s) { return stol(s); }
#define assert_yn(yn_v, v)                                                     \\
  ;                                                                            \\
  assert(yn_v == 0 || yn_v == v);                                              \\
  yn_v = v;
// 不完全な対策、現状はautohotkeyで対応
int yn_v = 0;
void yn(bool a) {
  assert_yn(yn_v, 1);
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void fyn(bool a) {
  assert_yn(yn_v, 1);
  yn(a);
  exit(0);
}

void Yn(bool a) {
  assert_yn(yn_v, 2);
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void fYn(bool a) {
  assert_yn(yn_v, 2);
  Yn(a);
  exit(0);
}

void YN(bool a) {
  assert_yn(yn_v, 3);
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fYN(bool a) {
  assert_yn(yn_v, 3);
  YN(a);
  exit(0);
}
int ab_v = 0;
void fAb(bool a) {
  assert_yn(ab_v, 1);
  if (a)
    cout << "Alice" << endl;
  else
    cout << "Bob";
}
void fAB(bool a) {
  assert_yn(yn_v, 2);
  if (a)
    cout << "ALICE" << endl;
  else
    cout << "BOB";
}
int pos_v = 0;
void Possible(bool a) {
  assert_yn(pos_v, 1);
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}
void POSSIBLE(bool a) {
  assert_yn(pos_v, 2);
  if (a)
    cout << "POSSIBLE" << endl;
  else
    cout << "IMPOSSIBLE" << endl;
  exit(0);
}
void fPossible(bool a) {
  assert_yn(pos_v, 1) Possible(a);
  exit(0);
}
void fPOSSIBLE(bool a) {
  assert_yn(pos_v, 2) POSSIBLE(a);
  exit(0);
}
template <typename T> class fixed_point : T {
public:
  explicit constexpr fixed_point(T &&t) noexcept : T(std::forward<T>(t)) {}
  template <typename... Args>
  constexpr decltype(auto) operator()(Args &&...args) const {
    return T::operator()(*this, std::forward<Args>(args)...);
  }
};
template <typename T>
static inline constexpr decltype(auto) fix(T &&t) noexcept {
  return fixed_point<T>{std::forward<T>(t)};
}

// 未分類
// 0,2,1 1番目と2番目の次元を入れ替える
template <class T>
auto irekae(vector<vector<vector<T>>> &A, int x, int y, int z) {
#define irekae_resize_loop(a, b, c)                                            \\
  resize(res, a, b, c);                                                        \\
  rep(i, a) rep(j, b) rep(k, c)
  vector<vector<vector<T>>> res;
  if (x == 0 && y == 1 && z == 2) {
    res = A;
  } else if (x == 0 && y == 2 && z == 1) {
    irekae_resize_loop(sz(A), sz(A[0][0]), sz(A[0])) {
      res[i][j][k] = A[i][k][j];
    }
  } else if (x == 1 && y == 0 && z == 2) {
    irekae_resize_loop(sz(A[0]), sz(A), sz(A[0][0])) {
      res[i][j][k] = A[j][i][k];
    }
  } else if (x == 1 && y == 2 && z == 0) {
    irekae_resize_loop(sz(A[0]), sz(A[0][0]), sz(A)) {
      res[i][j][k] = A[k][i][j];
    }
  } else if (x == 2 && y == 0 && z == 1) {
    irekae_resize_loop(sz(A[0][0]), sz(A), sz(A[0])) {
      res[i][j][k] = A[j][k][i];
    }
  } else if (x == 2 && y == 1 && z == 0) {
    irekae_resize_loop(sz(A[0][0]), sz(A[0]), sz(A)) {
      res[i][j][k] = A[k][j][i];
    }
  }
  return res;
#undef irekae_resize_loop
}
template <class T> auto irekae(vector<vector<T>> &A, int i = 1, int j = 0) {
  vvt(res, sz(A[0]), sz(A));
  rep(i, sz(A)) {
    rep(j, sz(A[0])) { res[j][i] = A[i][j]; }
  }
  return res;
}
// tou分割する
template <typename Iterable>
vector<Iterable> cut(const Iterable &a, int tou = 2) {
  int N = sz(a);
  vector<Iterable> res(tou);
  int hab = N / tou;
  vi lens(tou, hab);
  rep(i, N % tou) { lens[tou - 1 - i]++; }
  int l = 0;
  rep(i, tou) {
    int len = lens[i];
    int r = l + len;
    res[i].resize(len);
    std::copy(a.begin() + l, a.begin() + r, res[i].begin());
    l = r;
  }
  return res;
}
// 長さn毎に分割する
template <typename Iterable>
vector<Iterable> cut_n(const Iterable &a, int len) {
  int N = sz(a);
  vector<Iterable> res(ceil(N, len));
  vi lens(N / len, len);
  if (N % len)
    lens.push_back(N % len);
  int l = 0;
  rep(i, sz(lens)) {
    int len = lens[i];
    int r = l + len;
    res[i].resize(len);
    std::copy(a.begin() + l, a.begin() + r, res[i].begin());
    l = r;
  }
  return res;
}
// 縦を返す
vi &geth(vvi() & a, int w) {
  static vi ret;
  ret.resize(sz(a));
  rep(i, sz(a)) { ret[i] = a[i][w]; }
  return ret;
}

//@起動時
struct initon {
  initon() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout.setf(ios::fixed);
    cout.precision(16);
    srand((unsigned)clock() + (unsigned)time(NULL));
  };
} initonv;
// #define pre prev
// #define nex next

// gra mll pr
// 上下左右
const string udlr = "udlr";
string UDLR = "UDLR"; // x4と連動 UDLR.find('U') := x4[0]
vc atoz = {'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm',
           'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z'};
vc AtoZ = {'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M',
           'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z'};
// 右、上が正
constexpr ll h4[] = {1, -1, 0, 0};
constexpr ll w4[] = {0, 0, -1, 1};
constexpr ll h8[] = {0, 1, 0, -1, -1, 1, 1, -1};
constexpr ll w8[] = {1, 0, -1, 0, 1, -1, 1, -1};
int mei_inc(int h, int w, int H, int W, int i) {
  while (++i < 4) {
    if (inside(h + h4[i], w + w4[i], H, W))
      return i;
  }
  return i;
}
#define mei(nh, nw, h, w)                                                      \\
  for (int i = mei_inc(h, w, H, W, -1), nh = i < 4 ? h + h4[i] : 0,            \\
           nw = i < 4 ? w + w4[i] : 0;                                         \\
       i < 4; i = mei_inc(h, w, H, W, i), nh = h + h4[i], nw = w + w4[i])
int mei_inc8(int h, int w, int H, int W, int i) {
  while (++i < 8) {
    if (inside(h + h8[i], w + w8[i], H, W))
      return i;
  }
  return i;
}
#define mei8(nh, nw, h, w)                                                     \\
  for (int i = mei_inc8(h, w, H, W, -1), nh = i < 8 ? h + h8[i] : 0,           \\
           nw = i < 8 ? w + w8[i] : 0;                                         \\
       i < 8; i = mei_inc8(h, w, H, W, i), nh = h + h8[i], nw = w + w8[i])
int mei_incv(int h, int w, int H, int W, int i, vp &p) {
  while (++i < sz(p)) {
    if (inside(h + p[i].fi, w + p[i].se, H, W))
      return i;
  }
  return i;
}
#define meiv(nh, nw, h, w, p)                                                  \\
  for (int i = mei_incv(h, w, H, W, -1, p), nh = i < sz(p) ? h + p[i].fi : 0,  \\
           nw = i < sz(p) ? w + p[i].se : 0;                                   \\
       i < sz(p);                                                              \\
       i = mei_incv(h, w, H, W, i, p), nh = h + p[i].fi, nw = w + p[i].se)

// H*Wのグリッドを斜めに分割する
// 右上
vector<vp> naname_list_ne(int H, int W) {
  vector<vp> res(H + W - 1);
  rep(sh, H) {
    int sw = 0;
    res[sh] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh--;
      nw++;
      if (0 <= nh && nw < W) {
        res[sh] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  rep(sw, 1, W) {
    int sh = H - 1;
    res[H + sw - 1] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh--;
      nw++;
      if (0 <= nh && nw < W) {
        res[H + sw - 1] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  return res;
}
// 右下
vector<vp> naname_list_se(int H, int W) {
  vector<vp> res(H + W - 1);
  rep(sh, H) {
    int sw = 0;
    res[sh] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh++;
      nw++;
      if (0 <= nh && nh < H && nw < W) {
        res[sh] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  rep(sw, 1, W) {
    int sh = 0;
    res[H + sw - 1] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh++;
      nw++;
      if (0 <= nh && nh < H && nw < W) {
        res[H + sw - 1] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  return res;
}

// グラフ内で #undef getid
// #define getidとしているため、ここを書き直したらgraphも書き直す
#define getid_2(h, w) ((h) * (W) + (w))
#define getid_1(p) ((p).first * W + (p).second)
#define getid(...) over2(__VA_ARGS__, getid_2, getid_1)(__VA_ARGS__)
#define getp(id) mp((id) / W, (id) % W)
#define set_shuffle()                                                          \\
  std::random_device seed_gen;                                                 \\
  std::mt19937 engine(seed_gen())
#define shuffle(a) std::shuffle((a).begin(), (a).end(), engine);
// 1980 開始からtime ms経っていたらtrue
vb bit_bool(int v, int len) {
  assert(bit(len) > v);
  vb ret(len);
  rep(i, len) { ret[i] = bget(v, i); }
  return ret;
}

vi tov(vb &a) {
  vi ret;
  rep(i, sz(a)) {
    if (a[i])
      ret.push_back(i);
  }
  return ret;
}
bool kaibun(const str &S) { return S == rev(S); }
template <class T> vector<T> repeat(const vector<T> &A, int kaisu) {
  vector<T> ret;
  while (kaisu--) {
    ret += A;
  }
  return ret;
}
#define rge range
#define upd update
// S[{s, t, d}]
#define strs slice_str
struct slice_str {
  string S;
  slice_str() {}
  slice_str(const string &S) : S(S) {}
  slice_str(int len, char c) : S(len, c) {}
  auto size() { return S.size(); }
  char &operator[](int p) { return S[p]; }
  string operator[](initializer_list<int> p) {
    if (sz(p) == 1) {
      return S.substr(0, *(p.begin()));
    } else if (sz(p) == 2) {
      int l = *(p.begin());
      int r = *(next(p.begin()));
      return S.substr(l, r - l);
    } else {
      auto it = p.begin();
      int s = *(it++);
      int t = *(it++);
      int d = *(it);
      if (d == -1) {
        int s_ = sz(S) - s - 1;
        int t_ = sz(S) - t - 1;
        return rev(S).substr(s_, t_ - s_);
      } else if (d < 0) {
        t = max(-1ll, t);
        string ret;
        while (s > t) {
          ret += S[s];
          s += d;
        }
        return ret;
      } else {
        t = min(sz(S), t);
        string ret;
        while (s < t) {
          ret += S[s];
          s += d;
        }
        return ret;
      }
    }
  }
  operator string &() { return S; }
  template <class T> void operator+=(const T &a) { S += a; }
  bool operator==(const slice_str &rhs) { return S == rhs.S; }
  auto pop_back() { return S.pop_back(); }
};
ostream &operator<<(ostream &os, const slice_str &a) {
  os << a.S;
  return os;
}
istream &operator>>(istream &iss, slice_str &a) {
  iss >> a.S;
  return iss;
}
template <class T> bool can(const T &v, int i) { return 0 <= i && i < sz(v); }
#if __cplusplus >= 201703L
// template<class T> auto sum(int a, T  v...) {return (v + ... + 0);}
#endif
#define VEC vector

// string.substr()は長さを指定するため、間違えやすい
#define substr my_substr

template <class T> struct my_set : public set<T> {
  template <class... Arg> my_set(Arg... arg) : set<T>(arg...) {}
  T back() { return *(set<T>::rbegin()); }
  T front() { return *(set<T>::begin()); }
};
#define set my_set

#endif /*UNTITLED15_TEMPLATE_H*/

#endif

// † ←template終了
/*@formatter:on*/
// vectorで取れる要素数
// bool=> 1e9 * 8.32
// int => 1e8 * 2.6
// ll  => 1e8 * 1.3
// 3次元以上取るとメモリがヤバい
// static配列を使う
vvc(ba);
ll N, M, H, W;
vi A, B, C;
#ifndef MINT
#define MINT
/*@formatter:off*/
template <typename Mod_t> class modint;
template <typename T> T minv(T a, T m);
template <typename T> T minv(T a);

using Type = std::int_fast64_t;
constexpr signed MOD = 998244353;
//        1e9 + 7;//MOD

using mint = modint<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;

template <typename Mod_t> class modint {
  constexpr static auto mod() { return Mod_t::value; }

public:
  Type value;
  constexpr modint() : value() {}
  constexpr modint(const Type &x) noexcept {
    if (-mod() <= x && x < mod())
      value = static_cast<Type>(x);
    else
      value = static_cast<Type>(x % mod());
    if (value < 0)
      value += mod();
  }
  const Type &operator()() const { return value; }
  template <typename U> explicit operator U() const {
    return static_cast<U>(value);
  }
  constexpr modint operator+(const modint rhs) const noexcept {
    return modint(*this) += rhs;
  }
  constexpr modint operator-(const modint rhs) const noexcept {
    return modint(*this) -= rhs;
  }
  constexpr modint operator*(const modint rhs) const noexcept {
    return modint(*this) *= rhs;
  }
  constexpr modint operator/(const modint rhs) const noexcept {
    return modint(*this) /= rhs;
  }
  constexpr modint &operator+=(const modint rhs) noexcept {
    value += rhs.value;
    if (value >= mod()) {
      value -= mod();
    }
    return *this;
  }
  constexpr modint &operator-=(const modint rhs) noexcept {
    if (value < rhs.value) {
      value += mod();
    }
    value -= rhs.value;
    return *this;
  }
  constexpr modint &operator*=(const modint rhs) noexcept {
    value = value * rhs.value % mod();
    return *this;
  }
  constexpr modint &operator/=(const modint &rhs) noexcept {
    return operator*=(minv(rhs.value));
  }
  modint &operator++() { return *this += 1; }
  modint &operator--() { return *this -= 1; }
  modint operator++(signed) {
    modint result(*this);
    *this += 1;
    return result;
  }
  modint operator--(signed) {
    modint result(*this);
    *this -= 1;
    return result;
  }
  modint operator-() const { return modint(-value); }
  template <typename U>
  friend bool operator==(const modint<U> &lhs, const modint<U> &rhs);
  template <typename U>
  friend bool operator<(const modint<U> &lhs, const modint<U> &rhs);
  template <typename U>
  friend std::istream &operator>>(std::istream &stream, modint<U> &number);
  operator int() { return value; }
};
/*@formatter:off*/
template <typename T>
bool operator==(const modint<T> &lhs, const modint<T> &rhs) {
  return lhs.value == rhs.value;
}
template <typename T, typename U> bool operator==(const modint<T> &lhs, U rhs) {
  return lhs == modint<T>(rhs);
}
template <typename T, typename U> bool operator==(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) == rhs;
}
template <typename T>
bool operator!=(const modint<T> &lhs, const modint<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U> bool operator!=(const modint<T> &lhs, U rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U> bool operator!=(U lhs, const modint<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T>
bool operator<(const modint<T> &lhs, const modint<T> &rhs) {
  return lhs.value < rhs.value;
}
template <typename T>
modint<T> operator+(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T, typename U>
modint<T> operator+(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T, typename U>
modint<T> operator+(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T>
modint<T> operator-(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T, typename U>
modint<T> operator-(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T, typename U>
modint<T> operator-(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T>
modint<T> operator*(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T, typename U>
modint<T> operator*(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T, typename U>
modint<T> operator*(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T>
modint<T> operator/(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator/(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator/(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator%(const modint<T> &lhs, U rhs) {
  cerr << "can't  mint % int" << endl;
  assert(0);
  exit(0);
  return 0;
}

#ifdef optional
template <class T>
optional<mint> operator+(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv + opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator-(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator*(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator/(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv / opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator+(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv + opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator-(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator*(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator/(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv / opt.value();
  else
    return optional<mint>();
}
template <class T> void operator+=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv += opt.value();
}
template <class T> void operator-=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv -= opt.value();
}
template <class T> void operator*=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv *= opt.value();
}
template <class T> void operator/=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv /= opt.value();
}
template <class T> void operator+=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value += mv;
}
template <class T> void operator-=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value -= mv;
}
template <class T> void operator*=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value *= mv;
}
template <class T> void operator/=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value /= mv;
}
#endif
// constexpr int mint_len = std::min(1400001, MOD);
vector<Type> fac, finv, inv;
vector<Type> p2;
void setmod_resize(int mint_len) {
  //    p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i]
  //    = p2[i - 1] * 2 % MOD;
  int tn = sz(fac);
  fac.resize(mint_len);
  finv.resize(mint_len);
  inv.resize(mint_len);
  inv[1] = fac[0] = finv[0] = 1;
  for (int i = tn; i < mint_len; ++i)
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  for (int i = tn; i < mint_len; ++i)
    fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
}
// 4e6 >= n なら O(r)で計算する
mint com_big(int n, int r) {
  Type fa = finv[r];
  rep(i, r) fa *= n - i, fa %= MOD;
  return mint(fa);
}
mint com(int n, int r) {
  if (r < 0 || r > n)
    return 0; /*nが大きくてrが小さい場合、nを上からr個掛ける*/
  if (n >= sz(fac)) {
    if (n <= 4e6)
      setmod_resize(n + 1000);
    else
      return com_big(n, r);
  }
  return mint(finv[r] * fac[n] % MOD * finv[n - r]);
}
mint pom(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
mint npr(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
mint com_grid(int n, int r) {
  assert(r);
  return com(n + r - 2, n - 1);
}
Type nprin(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return fac[n] * finv[n - r] % MOD;
}
Type icom(int n, int r) {
  const int NUM_ = 1400001;
  static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];
  if (fac[0] == 0) {
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i <= NUM_; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i <= NUM_; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
  if (r < 0 || r > n)
    return 0;
  return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;
}
#define ncr com
#define ncri icom
// n種類でr個選ぶ
// n個の場所にr個の物を置く
// 負が渡された場合もcomが処理出来る
mint nhr(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return com(n + r - 1, r);
}
// n種類でそれぞれ少なくとも1つは選び、r個揃える
mint nhr1(int n, int r) { return nhr(n, r - n); }
mint hom(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return com(n + r - 1, r);
}
Type nhri(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return icom(n + r - 1, r);
}
// グリッドで0-indexedの最短経路 pascal
mint pas(int h, int w) { return com(h + w, w); }
template <typename T> T minv(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
template <typename T> T minv(T a) {
  if (a < sz(fac)) {
    return inv[a];
  }
  T u = 0, v = 1;
  T m = MOD;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
// int型 mod m
template <typename T, typename U> mint mpow(const T &a, const U &b) {
  if (b < 0)
    return minv(mpow(a, -b));
  mint x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      res *= x;
    x *= x;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
int mpowi(const T &a, const U &b, const V &m = MOD) {
  return mpow(a, b, m);
}
template <typename T> string to_string(const modint<T> &number) {
  return to_string(number());
}
/*@formatter:off*/
#ifdef _DEBUG
void yuri(const mint &a) {
  stringstream st;
  rep(i, 300) {
    rep(j, 300) {
      if ((mint)i / j == a) {
        st << i << " / " << j;
        i = 2000;
        break;
      }
    }
  }
  string val = st.str();
  if (val != "") {
    deb(val);
    return;
  }
  rep(i, 1000) {
    rep(j, 1000) {
      if ((mint)i / j == a) {
        st << i << " / " << j;
        i = 2000;
        break;
      }
    }
  }
  val = st.str();
  deb(val);
}
#else
#define yuri(...) ;
#endif
template <typename T>
std::ostream &operator<<(std::ostream &stream, const modint<T> &number) {
  stream << number();
#ifdef _DEBUG
//    stream << " -> " << yuri(number);
#endif
  return stream;
}
//@formatter:off
/*@formatter:on*/
template <typename T>
std::istream &operator>>(std::istream &stream, modint<T> &number) {
  typename common_type<typename modint<T>::Type, Type>::type x;
  stream >> x;
  number.value = modint<T>::normalize(x);
  return stream;
}
/*@formatter:off*/
using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define vvm(...)                                                               \\
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(mint, __VA_ARGS__)
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)
string out_m2(mint a) {
  stringstream st;
  st << (int)a;
  rep(i, 300) {
    rep(j, 2, 300) {
      if ((i % j) && (mint)i / j == a) {
        st << "(" << i << "/" << j << ")";
        i = 2000;
        break;
      }
    }
  }
  return st.str();
}
// nhr n個の場所にr個の物を分ける
mint m1 = (mint)1;
mint half = (mint)1 / 2;
struct setmod {
  // comで長さが足りない場合はsetmod_resizeでリサイズする
  int mint_len = 1.4 * 1e6;
  setmod() {
    // p2を使う場合setmod_resizeも変える
    //    p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i)
    //    p2[i] = p2[i - 1] * 2 % MOD;
    fac.resize(mint_len);
    finv.resize(mint_len);
    inv.resize(mint_len);
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i < mint_len; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i < mint_len; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
} setmodv;
template <signed S>
using mint_ = modint<std::integral_constant<decay<signed>::type, S>>;
#endif
/*@formatter:on*/
// 18:14
void solve() {
  din(sh, sw, th, tw);
  // 上が1, 2
  vnm(dp, th + 2, tw + 2, 3);
  dp[sh][sw][2] = 1;
  rep(h, sh, th + 1) {
    rep(w, sw, tw + 1) {
      rep(top, 1, 3) {
        // 右に追加
        // 上に二つ以上並ぶ必要がある
        if (top == 2) {
          dp[h][w + 1][2] += dp[h][w][top] * h;
        } else {
          // 一番上に追加しないといけない
          dp[h][w + 1][2] += dp[h][w][top];
        }

        // 上に追加
        dp[h + 1][w][1] += dp[h][w][top] * w;
      }
    }
  }
  out(dp[th][tw][1] + dp[th][tw][2]);
}

auto my(ll n, vi &a) { return 0; }

auto sister(ll n, vi &a) {
  ll ret = 0;
  return ret;
}

signed main() {
  solve();

#define arg n, a
#ifdef _DEBUG

  bool bad = 0;
  for (ll i = 0, ok = 1; i < k5 && ok; ++i) {
    ll n = rand(1, 8);
    vi a = ranv(n, 1, 10);
    auto myres = my(arg);
    auto res = sister(arg);
    ok = myres == res;
    if (!ok) {
      out(arg);
      cerr << "AC : " << res << endl;
      cerr << "MY  : " << myres << endl;
      bad = 1;

      break;
    }
  }
  if (!bad) {
    // solveを書き直す
    // solveを呼び出す
  }
  exit(0);
#endif
  return 0;
};


Error: WA

Fixed submission:

/*temp*/
//
//
//
//
// #undef _DEBUG
// #pragma GCC optimize("Ofast")
// 不動小数点の計算高速化
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
// #include <boost/multiprecision/cpp_int.hpp>

#ifdef _DEBUG
#include "template.h"
#else
#if __cplusplus >= 201703L
/*Atcoderでしか使えない(c++17 && このテンプレートが使えるならAtcoder)*/
#include <boost/sort/pdqsort/pdqsort.hpp>
#define fast_sort boost::sort::pdqsort
#endif
#endif

#ifndef _DEBUG
#ifndef UNTITLED15_TEMPLATE_H
#define UNTITLED15_TEMPLATE_H

#ifdef _DEBUG
#include "bits_stdc++.h"
#else
#include <bits/stdc++.h>
#endif

#ifndef fast_sort
#define fast_sort sort
#endif

// #define use_pq
#define use_for
#define use_for_each
#define use_sort
#define use_fill
#define use_rand
#define use_mgr
#define use_rui
#define use_compress
//
//
//
//
//
//
#define use_pbds
#ifdef use_pbds
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/tree_policy.hpp>

using namespace std;
using namespace std::chrono;

template <class T, class U>
void operator+=(__gnu_pbds::priority_queue<T, greater<T>,
                                           __gnu_pbds::rc_binomial_heap_tag> &a,
                const U &v) {
  a.push(v);
}
template <class T, class U>
void operator+=(
    __gnu_pbds::priority_queue<T, less<T>, __gnu_pbds::rc_binomial_heap_tag> &a,
    const U &v) {
  a.push(v);
}

template <class T, class U, class W, class X>
auto count(__gnu_pbds::gp_hash_table<T, U, W> &a, X k) {
  return a.find(k) != a.end();
}
#endif

/*@formatter:off*/
#define ll long long
using sig_dou = double;

// マクロ省略形 関数等
#define arsz(a) (sizeof(a) / sizeof(a[0]))
#define sz(a) ((ll)(a).size())
#define mp make_pair
#define mt make_tuple
#define pb pop_back
#define pf push_front
#define eb emplace_back
#define ALL(a) (a).begin(), (a).end()
#define RALL(a) (a).rbegin(), (a).rend()

template <class T, class U> auto max(T a, U b) { return a > b ? a : b; }
template <class T, class U> auto min(T a, U b) { return a < b ? a : b; }
// optional<T>について下でオーバーロード(nullopt_tを左辺右辺について単位元として扱う)
template <class T, class U> bool chma(T &a, const U &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <class T, class U> bool chmi(T &a, const U &b) {
  if (b < a) {
    a = b;
    return true;
  }
  return false;
}

// メタ系 meta
template <typename T, typename U = typename T::value_type>
std::true_type value_type_tester(signed);
template <typename T> std::false_type value_type_tester(long);

template <typename T>
struct has_value_type : decltype(value_type_tester<T>(0)) {};

// A<T>でTを返す
template <class T> using decl_t = typename T::value_type;
// A<B<.....T>>でTを返す
template <class T, bool end = !has_value_type<T>::value> struct decl2_ {
  typedef T type;
};
template <class T> struct decl2_<T, false> {
  typedef typename decl2_<decl_t<T>>::type type;
};
template <class T> using decl2_t = typename decl2_<T>::type;
// #define decl_max(a, b) decltype(max(MAX<decltype(a)>(), MAX<decltype(b)>()))
#define is_same2(T, U) is_same<T, U>::value
template <class T> struct is_vector : std::false_type {};
template <class T> struct is_vector<std::vector<T>> : std::true_type {};
// 大きい型を返す max_type<int, char>::type
// todo mintがlong long より小さいと判定されるためバグる
template <class T1, class T2, bool t1_bigger = (sizeof(T1) > sizeof(T2))>
struct max_type {
  typedef T1 type;
};
template <class T1, class T2> struct max_type<T1, T2, false> {
  typedef T2 type;
};

template <class T> struct vec_rank : integral_constant<int, 0> {};
template <class T>
struct vec_rank<vector<T>> : integral_constant<int, vec_rank<T>{} + 1> {};

// N個のTを並べたtupleを返す
// tuple_n<3, int>::type  tuple<int, int, int>
template <size_t N, class T, class... Arg> struct tuple_n {
  typedef typename tuple_n<N - 1, T, T, Arg...>::type type;
};
template <class T, class... Arg> struct tuple_n<0, T, Arg...> {
  typedef tuple<Arg...> type;
};

struct dummy_t1 {};
struct dummy_t2 {};
struct dummy_t3 {};
struct dummy_t4 {};
struct dummy_t5 {};
struct dummy_t6 {};
// template<class T, require(is_integral<T>::value)>など
#define require_t(bo) enable_if_t<bo> * = nullptr

// 複数でオーバーロードする場合、引数が同じだとうまくいかないため
// require_arg(bool, dummy_t1)
// require_arg(bool, dummy_t2)等とする
#define require_arg1(bo) enable_if_t<bo> * = nullptr
#define require_arg2(bo, dummy_type) enable_if_t<bo, dummy_type> * = nullptr
#define require_arg(...)                                                       \\
  over2(__VA_ARGS__, require_arg2, require_arg1)(__VA_ARGS__)
//->//enable_if_tのtを書き忘れそうだから
#define require_ret(bo, ret_type) enable_if_t<bo, ret_type>

#define int long long // todo 消したら動かない intの代わりにsignedを使う
auto start_time = system_clock::now();
auto past_time = system_clock::now();

#define debugName(VariableName) #VariableName
// 最大引数がN
#define over2(o1, o2, name, ...) name
#define over3(o1, o2, o3, name, ...) name
#define over4(o1, o2, o3, o4, name, ...) name
#define over5(o1, o2, o3, o4, o5, name, ...) name
#define over6(o1, o2, o3, o4, o5, o6, name, ...) name
#define over7(o1, o2, o3, o4, o5, o6, o7, name, ...) name
#define over8(o1, o2, o3, o4, o5, o6, o7, o8, name, ...) name
#define over9(o1, o2, o3, o4, o5, o6, o7, o8, o9, name, ...) name
#define over10(o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, name, ...) name

#ifdef _DEBUG
string message;
string res_mes;
bool was_deb = false;
[[noreturn]] void my_exit(signed e = 0) {
  if (was_deb && sz(res_mes)) {
    cerr << "result = " << endl << res_mes << endl;
  }
  if (sz(message)) {
    cerr << "****************************" << endl;
    cerr << "Note." << endl;
    cerr << message << endl;
    cerr << "****************************" << endl;
  }
  exit(e);
}
#define exit my_exit

#endif
void assert2(bool b, const string &s = "") {
  if (!b) {
    cerr << s << endl;
    exit(1); /*assert(0);*/
  }
}

// my_nulloptをあらゆる操作の単位元的な物として扱う
// vectorの参照外時に返したり、右辺値として渡されたときに何もしないなど
struct my_nullopt_t {
} my_nullopt;
#define nullopt_t my_nullopt_t
#define nullopt my_nullopt
/*@formatter:off*/
// 値が無いときは、setを使わない限り代入できない
//=を使っても無視される
template <class T> struct my_optional {
private:
  bool is_null;
  T v;

public:
  typedef T value_type;
  my_optional() : is_null(true) {}
  my_optional(const nullopt_t &) : is_null(true) {}
  my_optional(const T &v) : v(v), is_null(false) {}
  bool has_value() const { return !is_null; }
  T &value() {
    static string mes = "optional has no value";
    assert2(!is_null, mes);
    return v;
  }
  const T &value() const {
    static string mes = "optional has no value";
    assert2(!is_null, mes);
    return v;
  }
  void set(const T &nv) {
    is_null = false;
    v = nv;
  }
  template <class U> void operator=(const U &v) {
    set(v); // null状態でも代入出来るようにした
    //        if (has_value())value() = v; else return;
  }
  template <class U> void operator=(const my_optional<U> &v) {
    if (/*has_value() && */ v.has_value())
      (*this) = v;
    else
      return;
  }
  /*@formatter:off*/
  void reset() { is_null = true; }
  void operator=(const nullopt_t &) { reset(); }
  template <require_t(!is_same2(T, bool))> explicit operator bool() {
    return !is_null;
  }
  // nullの時はエラー
  operator T &() { return value(); }
  operator const T &() const { return value(); }
  my_optional<T> operator++() {
    if (this->has_value()) {
      this->value()++;
      return *this;
    } else {
      return *this;
    }
  }
  my_optional<T> operator++(signed) {
    if (this->has_value()) {
      auto tem = *this;
      this->value()++;
      return tem;
    } else {
      return *this;
    }
  }
  my_optional<T> operator--() {
    if (this->has_value()) {
      this->value()--;
      return *this;
    } else {
      return *this;
    }
  }
  my_optional<T> operator--(signed) {
    if (this->has_value()) {
      auto tem = *this;
      this->value()--;
      return tem;
    } else {
      return *this;
    }
  }
};
template <class T> istream &operator>>(istream &iss, my_optional<T> &v) {
  T val;
  iss >> val;
  v.set(val);
  return iss;
}

#define optional my_optional
template <class T> using opt = my_optional<T>;
// template<class T, class A = std::allocator<T>> struct debtor : std::vector<T,
// A> {
template <class Key, class T, class Compare = less<Key>,
          class Allocator = allocator<pair<const Key, T>>>
struct o_map : std::map<Key, optional<T>, Compare, Allocator> {
  optional<T> emp;
  o_map() : std::map<Key, optional<T>, Compare, Allocator>() {}
  auto operator()(const nullopt_t &) { return nullopt; }
  optional<T> &operator()(const optional<Key> &k) {
    if (k.has_value()) {
      return std::map<Key, optional<T>, Compare, Allocator>::operator[](
          k.value());
    } else {
      emp.reset();
      return emp;
    }
  }
  optional<T> &operator()(const Key &k) {
    auto &v = std::map<Key, optional<T>, Compare, Allocator>::operator[](k);
    if (v.has_value())
      return v;
    else {
      v.set(0);
      return v;
    }
  }
  template <class U> void operator[](U) {
    static string mes = "s_map cant []";
    assert2(0, mes);
  }
};
// 以下、空のoptionalをnulloptと書く
// ov[-1(参照外)] でnulloptを返す
// ov[nullopt] で nulloptをかえす

template <class T> struct ov {
  optional<T> emp;
  vector<optional<T>> v;
  ov(int i = 0, T val = 0) : v(i, val) {}
  template <class U> ov(const U &rhs) {
    v.resize(sz(rhs));
    for (int i = 0; i < sz(rhs); i++)
      v[i].set(rhs[i]);
  }
  optional<T> &operator()(int i) {
    if (i < 0 || sz(v) <= i) {
      emp.reset();
      return emp;
    } else {
      return v[i];
    }
  }
  optional<T> &operator()(const nullopt_t &) { return operator()(-1); }
  optional<T> &operator()(const optional<T> &i) {
    if (i.has_value())
      return operator()(i.value());
    else {
      return operator()(-1);
    }
  }
  /*@formatter:off*/
};
template <class T> string deb_tos(const ov<T> &v) { return deb_tos(v.v); }

// vectorに対しての処理は.vを呼ぶ
template <class T> class ovv {
  optional<T> emp;

public:
  vector<vector<optional<T>>> v;
  ovv(int i = 0, int j = 0, T val = 0) : v(i, vector<optional<T>>(j, val)) {}
  ovv(const vector<vector<T>> &S) {
    v.resize(S.size());
    for (int h = 0; h < sz(S); h++) {
      v[h].resize(sz(S[h]));
      for (int w = 0; w < sz(S[h]); w++) {
        v[h][w].set(S[h][w]);
      }
    }
  }
  optional<T> &operator()(int i, int j) {
    if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {
      emp.reset();
      return emp;
    } else {
      return v[i][j];
    }
  }
  // 再帰ver 遅いと思う
//    optional<T>& gets(optional<T>& v){return v;}
//    template<class V, class H, class... U> optional<T>& gets(V& v, H i, U...
//    tail){        if constexpr(is_same2(H, nullopt_t))return
//    operator()(-1,-1);        else if constexpr(is_same2(H, optional<int>)){
//    if(i.has_value())return gets(v[(int)i], tail...);            else return
//    operator()(-1,-1);        }else if constexpr(is_integral<H>::value){
//    return gets(v[(int)i], tail...);        }else{            assert(0);
//    return emp;        }    }
#if __cplusplus >= 201703L
  // if constexprバージョン 上が遅かったらこれで
  template <class U, class V>
  optional<T> &operator()(const U &i, const V &j) { /*駄目な場合を除外*/
    if constexpr (is_same2(U, nullopt_t) || is_same2(U, nullopt_t)) {
      return operator()(-1, -1); /* o, o*/
    } else if constexpr (is_same2(U, optional<int>) &&
                         is_same2(V, optional<int>)) {
      return operator()(i.has_value() ? (int)i : -1,
                        j.has_value() ? (int)j : -1); /* o, x*/
    } else if constexpr (is_same2(U, optional<int>)) {
      return operator()(i.has_value() ? (int)i : -1, (int)j); /* x, o*/
    } else if constexpr (is_same2(V, optional<int>)) {
      return operator()((int)i, j.has_value() ? (int)j : -1); /* x, x*/
    } else {
      return operator()((int)i, (int)j);
    }
  }
#endif
  operator const vector<vector<optional<T>>> &() { return v; }
};
template <class T> istream &operator>>(istream &iss, ovv<T> &a) {
  for (int h = 0; h < sz(a); h++) {
    for (int w = 0; w < sz(a[h]); w++) {
      iss >> a.v[h][w];
    }
  }
  return iss;
}
template <class T> ostream &operator<<(ostream &os, ovv<T> &a) {
  os << a.v;
  return os;
}
template <class T> string deb_tos(const ovv<T> &v) { return deb_tos(v.v); }
template <class T> struct ov3 {
  optional<T> emp;
  vector<vector<vector<optional<T>>>> v;
  ov3(int i, int j, int k, T val = 0)
      : v(i, vector<vector<optional<T>>>(j, vector<optional<T>>(k, val))) {}
  optional<T> &operator()(int i, int j, int k) {
    if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {
      if (k < 0 || sz(v[i][j]) <= k) {
        emp.reset();
        return emp;
      }
    }
    return v[i][j][k];
  }

private:
#if __cplusplus >= 201703L
  // 再帰ver 遅いと思う
  template <class V, class H> optional<T> &gets(V &nowv, H i) {
    if constexpr (is_same2(H, nullopt_t)) {
      emp.reset();
      return emp;
    } else if constexpr (is_same2(H, optional<int>)) {
      if (i.has_value()) {
        return nowv[(int)i];
      } else {
        emp.reset();
        return emp;
      }
    } else if constexpr (is_integral<H>::value) {
      return nowv[(int)i];
    } else {
      static string mes = "ov3 error not index";
      assert2(0, mes);
      emp.reset();
      return emp;
    }
  }
  // todo const &消した
  template <class V, class H, class... U>
  optional<T> &gets(V &nowv, H i, U... tail) {
    if constexpr (is_same2(H, nullopt_t)) {
      emp.reset();
      return emp;
    } else if constexpr (is_same2(H, optional<int>)) {
      if (i.has_value()) {
        return gets(nowv[(int)i], tail...);
      } else {
        emp.reset();
        return emp;
      }
    } else if constexpr (is_integral<H>::value) {
      return gets(nowv[(int)i], tail...);
    } else {
      static string mes = "ov3 error not index";
      assert2(0, mes);
      emp.reset();
      return emp;
    }
  }
#endif
public:
  template <class U, class V, class W> optional<T> &operator()(U i, V j, W k) {
    return gets(v, i, j, k);
  }
  /*@formatter:off*/
};
template <class T> string deb_tos(const ov3<T> &v) { return deb_tos(v.v); }
// nullopt_t
// 優先順位
// null, [opt, tem]

// + と += は違う意味を持つ
// val+=null : val
// val+null  : null
//
//+は途中計算
//+=は最終的に格納したい値にだけ持たせる
//+=がvoidを返すのは、途中計算で使うのを抑制するため

// nulloptを考慮する際、計算途中では+を使ってnulloptを作り
// 格納する際は+=で無効にする必要がある

// 演算子==
// optional<int>(10) == 10

// 全ての型に対応させ、value_typeが等しいかを見るようにするのもありかも

// null同士を比較する状況はおかしいのではないか
bool operator==(const nullopt_t &, const nullopt_t &) {
  assert2(0, "nul == null  cant hikaku");
  return false;
}
template <class T> bool operator==(const nullopt_t &, const T &) {
  return false;
}
template <class T> bool operator!=(const nullopt_t &, const T &) {
  return true;
}
template <class T> bool operator==(const T &, const nullopt_t &) {
  return false;
}
template <class T> bool operator!=(const T &, const nullopt_t &) {
  return true;
}
// nullを

nullopt_t &operator+(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator-(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator*(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator/(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator+=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator-=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator*=(const nullopt_t &, const nullopt_t &) { return nullopt; }
nullopt_t &operator/=(const nullopt_t &, const nullopt_t &) { return nullopt; }

template <class ANY> nullopt_t operator+(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator-(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator*(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator/(const nullopt_t &, const ANY &) {
  return nullopt;
}
template <class ANY> nullopt_t operator+(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator-(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator*(const ANY &, const nullopt_t &) {
  return nullopt;
}
template <class ANY> nullopt_t operator/(const ANY &, const nullopt_t &) {
  return nullopt;
}

template <class ANY> void operator+=(nullopt_t &, const ANY &) {}
template <class ANY> void operator-=(nullopt_t &, const ANY &) {}
template <class ANY> void operator*=(nullopt_t &, const ANY &) {}
template <class ANY> void operator/=(nullopt_t &, const ANY &) {}
template <class ANY> void operator+=(ANY &, const nullopt_t &) {}
template <class ANY> void operator-=(ANY &, const nullopt_t &) {}
template <class ANY> void operator*=(ANY &, const nullopt_t &) {}
template <class ANY> void operator/=(ANY &, const nullopt_t &) {}

template <class T> struct is_optional : false_type {};
template <class T> struct is_optional<optional<T>> : true_type {};
template <class T, class U>
true_type both_optional(optional<T> t, optional<U> u);
false_type both_optional(...);

template <class T, class U>
class opt_check : public decltype(both_optional(declval<T>(), declval<U>())) {};

// optionalは同じ型同士しか足せない
//(o, t), (t, o), (o, o)
#define opt_tem(op)                                                            \\
  template <class O, class O_ret = decltype(declval<O>() op declval<O>())>     \\
  optional<O_ret> operator op(const optional<O> &opt1,                         \\
                              const optional<O> &opt2) {                       \\
    if (!opt1.has_value() || !opt2.has_value()) {                              \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt1.value() op opt2.value());                    \\
    }                                                                          \\
  }                                                                            \\
  template <class O, class T,                                                  \\
            class O_ret = decltype(declval<O>() op declval<O>())>              \\
  auto operator op(const optional<O> &opt, const T &tem)                       \\
      ->require_ret(!(opt_check<optional<O>, T>::value), optional<O_ret>) {    \\
    if (!opt.has_value()) {                                                    \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt.value() op tem);                              \\
    }                                                                          \\
  }                                                                            \\
  template <class O, class T,                                                  \\
            class O_ret = decltype(declval<O>() op declval<O>())>              \\
  auto operator op(const T &tem, const optional<O> &opt)                       \\
      ->require_ret(!(opt_check<optional<O>, T>::value), optional<O_ret>) {    \\
    if (!opt.has_value()) {                                                    \\
      return optional<O_ret>();                                                \\
    } else {                                                                   \\
      return optional<O_ret>(opt.value() op tem);                              \\
    }                                                                          \\
  }
/*@formatter:off*/
opt_tem(+) opt_tem(-) opt_tem(*) opt_tem(/)
    // 比較はoptional<bool>を返す
    opt_tem(<) opt_tem(>) opt_tem(<=) opt_tem(>=)

    /*@formatter:on*/ /*@formatter:off*/

    template <class O, class T>
    bool operator==(const optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    return opt.value() == tem;
  } else
    return nullopt == tem;
}
template <class O, class T>
bool operator!=(const optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    return opt.value() != tem;
  } else
    return nullopt != tem;
}
template <class O, class T>
bool operator==(const T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    return opt.value() == tem;
  } else
    return nullopt == tem;
}
template <class O, class T>
bool operator!=(const T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    return opt.value() != tem;
  } else
    return nullopt != tem;
}

template <class O>
bool operator==(const optional<O> &opt1, const optional<O> &opt2) {
  if (opt1.has_value() != opt2.has_value()) {
    return false;
  } else if (opt1.has_value()) {
    return opt1.value() == opt2.value();
  } else {
    return nullopt == nullopt;
  }
}
template <class O>
bool operator!=(const optional<O> &opt1, const optional<O> &opt2) {
  return !(opt1 == opt2);
}

//(a+=null) != (a=a+null)
// a            null
template <class T, class O> void operator+=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem += opt.value();
  }
}
template <class T, class O> void operator-=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem -= opt.value();
  }
}
template <class T, class O> void operator*=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem *= opt.value();
  }
}
template <class T, class O> void operator/=(T &tem, const optional<O> &opt) {
  if (opt.has_value()) {
    tem /= opt.value();
  }
}

template <class T, class O> void operator+=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() += tem;
  }
}
template <class T, class O> void operator-=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() -= tem;
  }
}
template <class T, class O> void operator*=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() *= tem;
  }
}
template <class T, class O> void operator/=(optional<O> &opt, const T &tem) {
  if (opt.has_value()) {
    opt.value() /= tem;
  }
}
//
template <class Ol, class Or>
void operator+=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl += opr.value();
  }
}
template <class Ol, class Or>
void operator-=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl -= opr.value();
  }
}
template <class Ol, class Or>
void operator*=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl *= opr.value();
  }
}
template <class Ol, class Or>
void operator/=(optional<Ol> &opl, const optional<Or> &opr) {
  if (opr.has_value()) {
    return opl /= opr.value();
  }
}

/*@formatter:off*/

template <class U> auto max(const nullopt_t &, const U &val) { return val; }
template <class U> auto max(const U &val, const nullopt_t &) { return val; }
template <class U> auto min(const nullopt_t &, const U &val) { return val; }
template <class U> auto min(const U &val, const nullopt_t &) { return val; }

template <class T, class U> auto max(const optional<T> &opt, const U &val) {
  if (opt.has_value())
    return max(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto max(const U &val, const optional<T> &opt) {
  if (opt.has_value())
    return max(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto min(const optional<T> &opt, const U &val) {
  if (opt.has_value())
    return min(opt.value(), val);
  else
    return val;
}
template <class T, class U> auto min(const U &val, const optional<T> &opt) {
  if (opt.has_value())
    return min(opt.value(), val);
  else
    return val;
}

// null , optional, T
bool chma(nullopt_t &, const nullopt_t &) { return false; }
template <class T> bool chma(T &opt, const nullopt_t &) { return false; }
template <class T> bool chma(nullopt_t &, const T &opt) { return false; }
template <class T> bool chma(optional<T> &olv, const optional<T> &orv) {
  if (orv.has_value()) {
    return chma(olv, orv.value());
  } else
    return false;
}
template <class T, class U> bool chma(optional<T> &opt, const U &rhs) {
  if (opt.has_value()) {
    return chma(opt.value(), rhs);
  } else
    return false;
}
template <class T, class U> bool chma(T &lhs, const optional<U> &opt) {
  if (opt.has_value()) {
    return chma(lhs, opt.value());
  } else
    return false;
}

bool chmi(nullopt_t &, const nullopt_t &) { return false; }
template <class T> bool chmi(T &opt, const nullopt_t &) { return false; }
template <class T> bool chmi(nullopt_t &, const T &opt) { return false; }
template <class T> bool chmi(optional<T> &olv, const optional<T> &orv) {
  if (orv.has_value()) {
    return chmi(olv, orv.value());
  } else
    return false;
}
template <class T, class U> bool chmi(optional<T> &opt, const U &rhs) {
  if (opt.has_value()) {
    return chmi(opt.value(), rhs);
  } else
    return false;
}
template <class T, class U> bool chmi(T &lhs, const optional<U> &opt) {
  if (opt.has_value()) {
    return chmi(lhs, opt.value());
  } else
    return false;
}

template <class T> ostream &operator<<(ostream &os, optional<T> p) {
  if (p.has_value())
    os << p.value();
  else
    os << "e";
  return os;
}
template <class T> using opt = my_optional<T>;
struct xorshift {
  /*@formatter:on*/
  static uint64_t splitmix64(uint64_t x) {
    x += 0x9e3779b97f4a7c15;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
  }
  /*@formatter:off*/
  size_t operator()(const uint64_t &x) const {
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    return splitmix64(x + FIXED_RANDOM);
  }
  size_t operator()(const std::pair<ll, ll> &x) const {
    ll v = ((x.first) << 32) | x.second;
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    return splitmix64(v + FIXED_RANDOM);
  }
  template <class T, class U>
  size_t operator()(const std::pair<T, U> &x) const {
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    uint64_t hasx = splitmix64(x.first);
    uint64_t hasy = splitmix64(x.second + FIXED_RANDOM);
    return hasx ^ hasy;
  }
  template <class T> size_t operator()(const vector<T> &x) const {
    uint64_t has = 0;
    static const uint64_t FIXED_RANDOM =
        chrono::steady_clock::now().time_since_epoch().count();
    uint64_t rv = splitmix64(FIXED_RANDOM);
    for (int i = 0; i < sz(x); i++) {
      uint64_t v = splitmix64(x[i] + rv);
      has ^= v;
      rv = splitmix64(rv);
    }
    return has;
  }
};

#ifdef _DEBUG
// #define use_debtor
// template<class T, class U, class X> auto count(unordered_map<T, U> &a, X k) {
// return a.find(k) != a.end(); }

#ifdef use_debtor
// https://marycore.jp/prog/cpp/class-extension-methods/ 違うかも
template <class T, class A = std::allocator<T>>
struct debtor : std::vector<T, A> {
  using std::vector<T, A>::vector;
  template <class U> int deb_v(U a, int v) { return v; }
  template <class U> int deb_v(debtor<U> &a, int v = 0) {
    cerr << a.size() << " ";
    return deb_v(a.at(0), v + 1);
  }
  template <class U> void deb_o(U a) { cerr << a << " "; }
  template <class U> void deb_o(debtor<U> &a) {
    for (int i = 0; i < min((int)a.size(), 15ll); i++) {
      deb_o(a[i]);
    }
    if ((int)a.size() > 15) {
      cerr << "...";
    }
    cerr << endl;
  }
  typename std::vector<T>::reference my_at(typename std::vector<T>::size_type n,
                                           vector<int> &ind) {
    if (n < 0 || n >= (int)this->size()) {
      int siz = (int)this->size();
      cerr << "vector size = ";
      int dim = deb_v((*this));
      cerr << endl;
      ind.push_back(n);
      cerr << "out index at  ";
      for (auto &&i : ind) {
        cerr << i << " ";
      }
      cerr << endl;
      cerr << endl;
      if (dim <= 2) {
        deb_o((*this));
      }
      exit(0);
    }
    return this->at(n);
  }
  typename std::vector<T>::reference
  operator[](typename std::vector<T>::size_type n) {
    if (n < 0 || n >= (int)this->size()) {
      int siz = (int)this->size();
      cerr << "vector size = ";
      int dim = deb_v((*this));
      cerr << endl;
      cerr << "out index at  " << n << endl;
      cerr << endl;
      if (dim <= 2) {
        deb_o((*this));
      }
      exit(0);
    }
    return this->at(n);
  }
};
#define vector debtor
#endif
#ifdef use_pbds
template <class T> struct my_pbds_tree {
  set<T> s;
  auto begin() { return s.begin(); }
  auto end() { return s.end(); }
  auto rbegin() { return s.rbegin(); }
  auto rend() { return s.rend(); }
  auto empty() { return s.empty(); }
  auto size() { return s.size(); }
  void clear() { s.clear(); }
  template <class U> void insert(U v) { s.insert(v); }
  template <class U> void operator+=(U v) { insert(v); }
  template <class F> auto erase(F v) { return s.erase(v); }
  template <class U> auto find(U v) { return s.find(v); }
  template <class U> auto lower_bound(U v) { return s.lower_bound(v); }
  template <class U> auto upper_bound(U v) { return s.upper_bound(v); }
  auto find_by_order(ll k) {
    auto it = s.begin();
    for (ll i = 0; i < k; i++)
      it++;
    return it;
  }
  auto order_of_key(ll v) {
    auto it = s.begin();
    ll i = 0;
    for (; it != s.end() && *it < v; i++)
      it++;
    return i;
  }
};
#define pbds(T) my_pbds_tree<T>
#endif

// 区間削除は出来ない
// gp_hash_tableでcountを使えないようにするため
template <class K, class V> using umap_f = unordered_map<K, V>;
template <typename K, typename T> using map_f = map<K, T>;

#else
#define endl '\\n'
// umapはunorderd_mapになる
// umapiはgp_hash_table

// find_by_order(k) k番目のイテレーター
// order_of_key(k)  k以上が前から何番目か
#define pbds(U)                                                                \\
  __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, \\
                   __gnu_pbds::tree_order_statistics_node_update>

template <class K, class V>
struct umap_f : public __gnu_pbds::gp_hash_table<K, V, xorshift> {
  int count(const K &k) { return this->find(k) != this->end(); }
};
template <typename K, typename T>
using map_f = __gnu_pbds::tree<K, T, less<K>, __gnu_pbds::rb_tree_tag,
                               __gnu_pbds::tree_order_statistics_node_update>;

#endif
#define umapi unordered_map<ll, ll>
#define umapp unordered_map<P, ll>
#define umappp unordered_map<P, P>
#define umapu unordered_map<uint64_t, ll>
#define umapip unordered_map<ll, P>

template <class T, class U, class X> auto count(unordered_map<T, U> &a, X k) {
  return a.find(k) != a.end();
}
/*@formatter:off*/
#ifdef use_pbds
template <class U, class L>
void operator+=(
    __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag,
                     __gnu_pbds::tree_order_statistics_node_update> &s,
    L v) {
  s.insert(v);
}
#endif
// 衝突対策
#define ws ws_

typedef tuple<ll, ll, ll> T;
typedef tuple<ll, ll, ll, ll> F;
// T mt(ll a, ll b, ll c) { return T(a, b, c); }
// F mf(ll a, ll b, ll c, ll d) { return F(a, b, c, d); }

// 関数内をまとめる
// 初期値l=-1, r=-1
void set_lr12(int &l, int &r, int n) { /*r==-1*/
  if (r == -1) {
    if (l == -1) {
      l = 0;
      r = n;
    } else {
      r = l;
      l = 0;
    }
  }
}
//@マクロ省略系 型,構造
// using で元のdoubleを同時に使えるはず
#define double_big

#ifdef double_big
#define double long double
// #define pow powl
#endif
using dou = double;

/*@formatter:off*/
template <class T> T MAX() { return numeric_limits<T>::max(); }
template <class T> T MIN() { return numeric_limits<T>::min(); }
constexpr ll inf = (ll)1e9 + 100;
constexpr ll linf = (ll)1e18 + 100;
constexpr dou dinf = (dou)linf * linf;
constexpr char infc = '{';
const string infs = "{";

template <class T> T INF() { return MAX<T>() / 2; }
template <> signed INF() { return inf; }
template <> ll INF() { return linf; }
template <> double INF() { return dinf; }
template <> char INF() { return infc; }
template <> string INF() { return infs; }

const double eps = 1e-9;

// #define use_epsdou
#ifdef use_epsdou
// 基本コメントアウト
struct epsdou {
  double v;
  epsdou(double v = 0) : v(v) {}
  template <class T> epsdou &operator+=(T b) {
    v += (double)b;
    return (*this);
  }
  template <class T> epsdou &operator-=(T b) {
    v -= (double)b;
    return (*this);
  }
  template <class T> epsdou &operator*=(T b) {
    v *= (double)b;
    return (*this);
  }
  template <class T> epsdou &operator/=(T b) {
    v /= (double)b;
    return (*this);
  }
  epsdou operator+(epsdou b) { return v + (double)b; }
  epsdou operator-(epsdou b) { return v - (double)b; }
  epsdou operator*(epsdou b) { return v * (double)b; }
  epsdou operator/(epsdou b) { return v / (double)b; }
  epsdou operator-() const { return epsdou(-v); }
  template <class T> bool operator<(T b) { return v < (double)b; }
  template <class T> bool operator>(T b) {
    auto r = (double)b;
    return v > (double)b;
  }
  template <class T> bool operator==(T b) { return fabs(v - (double)b) <= eps; }
  template <class T> bool operator<=(T b) {
    return v < (double)b || fabs(v - b) <= eps;
  }
  template <class T> bool operator>=(T b) {
    return v > (double)b || fabs(v - b) <= eps;
  }
  operator double() { return v; }
};
template <> epsdou MAX() { return MAX<double>(); }
template <> epsdou MIN() { return MIN<double>(); }
// priqrity_queue等で使うのに必要
bool operator<(const epsdou &a, const epsdou &b) { return a.v < b.v; }
bool operator>(const epsdou &a, const epsdou &b) { return a.v > b.v; }
istream &operator>>(istream &iss, epsdou &a) {
  iss >> a.v;
  return iss;
}
ostream &operator<<(ostream &os, epsdou &a) {
  os << a.v;
  return os;
}
#define eps_conr_t(o)                                                          \\
  template <class T> epsdou operator o(T a, epsdou b) { return (dou)a o b.v; }
#define eps_conl_t(o)                                                          \\
  template <class T> epsdou operator o(epsdou a, T b) { return a.v o(dou) b; }
eps_conl_t(+) eps_conl_t(-) eps_conl_t(*) eps_conl_t(/) eps_conr_t(+)
    eps_conr_t(-) eps_conr_t(*) eps_conr_t(/)
// template<class U> epsdou max(epsdou a, U b){return a.v>b ? a.v: b;}
// template<class U> epsdou max(U a, epsdou b){return a>b.v ? a: b.v;}
// template<class U> epsdou min(epsdou a, U b){return a.v<b ? a.v: b;}
// template<class U> epsdou min(U a, epsdou b){return a<b.v ? a: b.v;}
#undef double
#define double epsdou
#undef dou
#define dou epsdou
#endif

        template <class T = int, class A, class B = int>
        T my_pow(A a, B b = 2) {
  if (b < 0)
    return (T)1 / my_pow<T>(a, -b);
#if __cplusplus >= 201703L
  if constexpr (is_floating_point<T>::value) {
    return pow((T)a, (T)b);
  } else if constexpr (is_floating_point<A>::value) {
    assert2(
        0,
        "pow <not dou>(dou, )"); /*return
                                    0;しない方がコンパイル前に(voidを受け取るので)エラーが出ていいかも*/
  } else if constexpr (is_floating_point<B>::value) {
    assert2(
        0,
        "pow <not dou>(, dou)"); /*return
                                    0;しない方がコンパイル前に(voidを受け取るので)エラーが出ていいかも*/
  } else {
#endif
    T ret = 1;
    T bek = a;
    while (b) {
      if (b & 1)
        ret *= bek;
      bek *= bek;
      b >>= 1;
    }
    return ret;
#if __cplusplus >= 201703L
  }
#endif
}
#define pow my_pow

#define ull unsigned long long
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;

#define fi first
#define se second
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define el else
#define elf else if
#define upd update
#define sstream stringstream

#define maxq 1
#define minq -1

#define ZERO(a) memset(a, 0, sizeof(a))
#define MINUS(a) memset(a, 0xff, sizeof(a))
#define MALLOC(type, len) (type *)malloc((len) * sizeof(type))
#define lam1(ret) [&](auto &&v) { return ret; }
#define lam2(v, ret) [&](auto &&v) { return ret; }
#define lam(...) over2(__VA_ARGS__, lam2, lam1)(__VA_ARGS__)
#define lamr(right) [&](auto &&p) { return p right; }
#define unique(v) v.erase(unique(v.begin(), v.end()), v.end());

// マクロ省略系 コンテナ
using vi = vector<ll>;
using vb = vector<bool>;
using vs = vector<string>;
using vd = vector<double>;
using vc = vector<char>;
using vp = vector<P>;
using vt = vector<T>;

// #define V vector
#define vvt0(t) vector<vector<t>>
#define vvt1(t, a) vector<vector<t>> a
#define vvt2(t, a, b) vector<vector<t>> a(b)
#define vvt3(t, a, b, c) vector<vector<t>> a(b, vector<t>(c))
#define vvt4(t, a, b, c, d) vector<vector<t>> a(b, vector<t>(c, d))

#define vv(type, ...)                                                          \\
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(type, __VA_ARGS__)
#define vvi(...) vv(ll, __VA_ARGS__)
#define vvb(...) vv(bool, __VA_ARGS__)
#define vvs(...) vv(string, __VA_ARGS__)
#define vvd(...) vv(double, __VA_ARGS__)
#define vvc(...) vv(char, __VA_ARGS__)
#define vvp(...) vv(P, __VA_ARGS__)
#define vvt(...) vv(T, __VA_ARGS__)
// optional
#define vvoi(...) vv(optional<ll>, __VA_ARGS__)

template <typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template <typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
#define vni(name, ...) auto name = make_v<ll>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)
#define vn(type, name, ...) auto name = make_v<type>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll>>
#define tos to_string
using mapi = map<ll, ll>;
using mapp = map<P, ll>;
using mapd = map<dou, ll>;
using mapc = map<char, ll>;
using maps = map<str, ll>;
using seti = set<ll>;
using setp = set<P>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<ll>;
#define uset unordered_set
#define useti unordered_set<ll, xorshift>
#define mset multiset
#define mseti multiset<ll>
#define umap unordered_map
#define mmap multimap

// 初期化子を与える
template <template <class...> class MAP, class K, class V, class... A>
struct init_map : public MAP<K, V, A...> {
  V init_value;
  init_map(V init_value = 0) : MAP<K, V, A...>(), init_value(init_value) {}
  V &operator[](const K &k) {
    if (MAP<K, V, A...>::count(k) == 0) {
      return MAP<K, V, A...>::operator[](k) = init_value;
    }
    return MAP<K, V, A...>::operator[](k);
  }
};
template <class... A> using map2 = init_map<map, A...>;
template <class... A> using umap2 = init_map<umap, A...>;
template <class... A> using umap_f2 = init_map<umap_f, A...>;

// 任意のマクロサポート用 使う度に初期化する //todo これを消す
int index_, v1_, v2_, v3_;

/*@formatter:off*/
string to_string(char c) {
  string ret = "";
  ret += c;
  return ret;
}
template <class T> class pq_min_max {
  vector<T> d;
  void make_heap() {
    for (int i = d.size(); i--;) {
      if (i & 1 && d[i - 1] < d[i])
        swap(d[i - 1], d[i]);
      int k = down(i);
      up(k, i);
    }
  }
  inline int parent(int k) const { return ((k >> 1) - 1) & ~1; }
  int down(int k) {
    int n = d.size();
    if (k & 1) { /* min heap*/
      while (2 * k + 1 < n) {
        int c = 2 * k + 3;
        if (n <= c || d[c - 2] < d[c])
          c -= 2;
        if (c < n && d[c] < d[k]) {
          swap(d[k], d[c]);
          k = c;
        } else
          break;
      }
    } else { /* max heap*/
      while (2 * k + 2 < n) {
        int c = 2 * k + 4;
        if (n <= c || d[c] < d[c - 2])
          c -= 2;
        if (c < n && d[k] < d[c]) {
          swap(d[k], d[c]);
          k = c;
        } else
          break;
      }
    }
    return k;
  }
  int up(int k, int root = 1) {
    if ((k | 1) < (int)d.size() && d[k & ~1] < d[k | 1]) {
      swap(d[k & ~1], d[k | 1]);
      k ^= 1;
    }
    int p;
    while (root < k && d[p = parent(k)] < d[k]) { /*max heap*/
      swap(d[p], d[k]);
      k = p;
    }
    while (root < k && d[k] < d[p = parent(k) | 1]) { /* min heap*/
      swap(d[p], d[k]);
      k = p;
    }
    return k;
  }

public:
  pq_min_max() {}
  pq_min_max(const vector<T> &d_) : d(d_) { make_heap(); }
  template <class Iter> pq_min_max(Iter first, Iter last) : d(first, last) {
    make_heap();
  }
  void operator+=(const T &x) {
    int k = d.size();
    d.push_back(x);
    up(k);
  }
  void pop_min() {
    if (d.size() < 3u) {
      d.pop_back();
    } else {
      swap(d[1], d.back());
      d.pop_back();
      int k = down(1);
      up(k);
    }
  }
  void pop_max() {
    if (d.size() < 2u) {
      d.pop_back();
    } else {
      swap(d[0], d.back());
      d.pop_back();
      int k = down(0);
      up(k);
    }
  }
  const T &get_min() const { return d.size() < 2u ? d[0] : d[1]; }
  const T &get_max() const { return d[0]; }
  int size() const { return d.size(); }
  bool empty() const { return d.empty(); }
};
// 小さいほうからM個取得するpq
template <class T> struct helper_pq_size {
  pq_min_max<T> q;
  T su = 0;
  int max_size = 0;
  helper_pq_size() {}
  helper_pq_size(int max_size) : max_size(max_size) {}
  void clear() {
    q = pq_min_max<T>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q += (v);
    if (sz(q) > max_size) {
      su -= q.get_max();
      q.pop_max();
    }
  }
  T sum() { return su; }
  T top() { return q.get_min(); }
  void pop() {
    su -= q.get_min();
    q.pop_min();
  }
  T poll() {
    T ret = q.get_min();
    su -= ret;
    q.pop_min();
    return ret;
  }
  ll size() { return q.size(); }
};
// 大きいほうからM個取得するpq
template <class T> struct helper_pqg_size {
  pq_min_max<T> q;
  T su = 0;
  int max_size = 0;
  helper_pqg_size() {}
  helper_pqg_size(int max_size) : max_size(max_size) {}
  void clear() {
    q = pq_min_max<T>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q += (v);
    if (sz(q) > max_size) {
      su -= q.get_min();
      q.pop_min();
    }
  }
  T sum() { return su; }
  T top() { return q.get_max(); }
  void pop() {
    su -= q.get_max();
    q.pop_max();
  }
  T poll() {
    T ret = q.get_max();
    su -= ret;
    q.pop_min();
    return ret;
  }
  ll size() { return q.size(); }
};
;

template <class T, class Container = vector<T>,
          class Compare = std::less<typename Container::value_type>>
struct helper_pqg {
  priority_queue<T, Container, Compare> q; /*小さい順*/
  T su = 0;
  helper_pqg() {}
  void clear() {
    q = priority_queue<T, vector<T>, greater<T>>();
    su = 0;
  }
  void operator+=(T v) {
    su += v;
    q.push(v);
  }
  T sum() { return su; }
  T top() { return q.top(); }
  void pop() {
    su -= q.top();
    q.pop();
  }
  T poll() {
    T ret = q.top();
    su -= ret;
    q.pop();
    return ret;
  }
  ll size() { return q.size(); }
};
template <class T> using helper_pq = helper_pqg<T, vector<T>, greater<T>>;
#if __cplusplus >= 201703L

// 小さいほうからsize個残る
// Tがoptionalなら空の時nullを返す
template <class T> struct pq {
  helper_pq<T> a_q;      /*大きい順*/
  helper_pq_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pq(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pq_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  // optionalなら空の時nullを返す
  T top() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q) == 0)
          return T();
        return a_q.top();
      } else {
        if (sz(b_q) == 0)
          return T();
        return b_q.top();
      }
    } else {
      if (aquery)
        return a_q.top();
      else
        return b_q.top();
    }
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  // optionalなら空の時何もしない
  void pop() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q))
          a_q.pop();
      } else {
        if (sz(b_q))
          b_q.pop();
      }
    } else {
      if (aquery)
        a_q.pop();
      else
        b_q.pop();
    }
  } /*T*/
  T poll() {
    if constexpr (is_optional<T>::value) {
      if (aquery) {
        if (sz(a_q) == 0)
          return T();
        return a_q.poll();
      } else {
        if (sz(b_q) == 0)
          return T();
        return b_q.poll();
      }
    } else {
      if (aquery)
        return a_q.poll();
      else
        return b_q.poll();
    }
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
  /*@formatter:off*/
};
template <class T> struct pqg {
  helper_pqg<T> a_q;      /*大きい順*/
  helper_pqg_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pqg(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pqg_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
#else
// 小さいほうからsize個残る
template <class T> struct pq {
  helper_pq<T> a_q;      /*大きい順*/
  helper_pq_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pq(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pq_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
// 大きいほうからsize個残る
template <class T> struct pqg {
  helper_pqg<T> a_q;      /*大きい順*/
  helper_pqg_size<T> b_q; /*大きい順*/
  bool aquery;
  T su = 0;
  pqg(int size = inf) {
    aquery = size == inf;
    if (!aquery) {
      b_q = helper_pqg_size<T>(size);
    }
  }
  void clear() {
    if (aquery)
      a_q.clear();
    else
      b_q.clear();
  }
  void operator+=(T v) {
    if (aquery)
      a_q += v;
    else
      b_q += v;
  }
  T sum() {
    if (aquery)
      return a_q.sum();
    else
      return b_q.sum();
  }
  T top() {
    if (aquery)
      return a_q.top();
    else
      return b_q.top();
  }
  void pop() {
    if (aquery)
      a_q.pop();
    else
      b_q.pop();
  }
  T poll() {
    if (aquery)
      return a_q.poll();
    else
      return b_q.poll();
  }
  ll size() {
    if (aquery)
      return a_q.size();
    else
      return b_q.size();
  }
};
#endif
#define pqi pq<ll>
#define pqgi pqg<ll>
template <class T> string deb_tos(pq<T> &q) {
  vector<T> res;
  auto temq = q;
  while (sz(temq))
    res.push_back(temq.top()), temq.pop();
  stringstream ss;
  ss << res;
  return ss.str();
}
template <class T> string deb_tos(pqg<T> &q) {
  vector<T> res;
  auto temq = q;
  while (sz(temq))
    res.push_back(temq.top()), temq.pop();
  stringstream ss;
  ss << res;
  return ss.str();
}

/*@formatter:off*/
// マクロ 繰り返し
// ↓@オーバーロード隔離
// todo 使わないもの非表示
#define rep1(n) for (ll rep1i = 0, rep1lim = n; rep1i < rep1lim; ++rep1i)
#define rep2(i, n) for (ll i = 0, rep2lim = n; i < rep2lim; ++i)
#define rep3(i, m, n) for (ll i = m, rep3lim = n; i < rep3lim; ++i)
#define rep4(i, m, n, ad) for (ll i = m, rep4lim = n; i < rep4lim; i += ad)
// 逆順 閉区間
#define rer2(i, n) for (ll i = n; i >= 0; i--)
#define rer3(i, m, n) for (ll i = m, rer3lim = n; i >= rer3lim; i--)
#define rer4(i, m, n, dec) for (ll i = m, rer4lim = n; i >= rer4lim; i -= dec)

#ifdef use_for

// ループを一つにまとめないとフォーマットで汚くなるため
#define nex_ind1(i) i++
#define nex_ind2(i, j, J)                                                      \\
  i = (j + 1 == J) ? i + 1 : i, j = (j + 1 == J ? 0 : j + 1)
#define nex_ind3(i, j, k, J, K)                                                \\
  i = (j + 1 == J && k + 1 == K) ? i + 1 : i,                                  \\
  j = (k + 1 == K) ? (j + 1 == J ? 0 : j + 1) : j,                             \\
  k = (k + 1 == K ? 0 : k + 1)
#define nex_ind4(i, j, k, l, J, K, L)                                          \\
  i = (j + 1 == J && k + 1 == K && l + 1 == L) ? i + 1 : i,                    \\
  j = (k + 1 == K && l + 1 == L) ? (j + 1 == J ? 0 : j + 1) : j,               \\
  k = (l + 1 == L ? (k + 1 == K ? 0 : k + 1) : k), l = l + 1 == L ? 0 : l + 1
#define nex_ind5(i, j, k, l, m, J, K, L, M)                                    \\
  i = (j + 1 == J && k + 1 == K && l + 1 == L && m + 1 == M) ? i + 1 : i,      \\
  j = (k + 1 == K && l + 1 == L && m + 1 == M) ? (j + 1 == J ? 0 : j + 1) : j, \\
  k = (l + 1 == L && m + 1 == M ? (k + 1 == K ? 0 : k + 1) : k),               \\
  l = m + 1 == M ? l + 1 == L ? 0 : l + 1 : l, m = m + 1 == M ? 0 : m + 1

#define repss2(i, I) for (int i = 0; i < I; i++)
#define repss4(i, j, I, J)                                                     \\
  for (int i = (J ? 0 : I), j = 0; i < I; nex_ind2(i, j, J))
#define repss6(i, j, k, I, J, K)                                               \\
  for (int i = (J && K ? 0 : I), j = 0, k = 0; i < I; nex_ind3(i, j, k, J, K))
#define repss8(i, j, k, l, I, J, K, L)                                         \\
  for (int i = (J && K && L ? 0 : I), j = 0, k = 0, l = 0; i < I;              \\
       nex_ind4(i, j, k, l, J, K, L))
#define repss10(i, j, k, l, m, I, J, K, L, M)                                  \\
  for (int i = (J && K && L && M ? 0 : I), j = 0, k = 0, l = 0, m = 0; i < I;  \\
       nex_ind5(i, j, k, l, m, J, K, L, M))

// i,j,k...をnまで見る
#define reps2(i, n) repss2(i, n)
#define reps3(i, j, n) repss4(i, j, n, n)
#define reps4(i, j, k, n) repss6(i, j, k, n, n, n)
#define reps5(i, j, k, l, n) repss8(i, j, k, l, n, n, n, n)
template <class T>
void nex_repv2(int &i, int &j, int &I, int &J, vector<vector<T>> &s) {
  while (1) {
    j++;
    if (j >= J) {
      j = 0;
      i++;
      if (i < I) {
        J = (int)s[i].size();
      }
    }
    if (i >= I || J)
      return;
  }
}
template <class T>
void nex_repv3(int &i, int &j, int &k, int &I, int &J, int &K,
               vector<vector<vector<T>>> &s) {
  while (1) {
    k++;
    if (k >= K) {
      k = 0;
      j++;
      if (j >= J) {
        j = 0;
        i++;
        if (i >= I)
          return;
      }
    }
    J = (int)s[i].size();
    K = (int)s[i][j].size();
    if (J && K)
      return;
  }
}

#define repv_2(i, a) repss2(i, sz(a))
// 正方形である必要はない
// 直前を持つのとどっちが早いか
#define repv_3(i, j, a)                                                        \\
  for (int repvI = (int)a.size(), repvJ = (int)a[0].size(), i = 0, j = 0;      \\
       i < repvI; nex_repv2(i, j, repvI, repvJ, a))
// 箱状になっている事が要求される つまり[i] 次元目の要素数は一定
#define repv_4(i, j, k, a)                                                     \\
  for (int repvI = (int)a.size(), repvJ = (int)a[0].size(),                    \\
           repvK = (int)a[0][0].size(), i = 0, j = 0, k = 0;                   \\
       i < repvI; nex_repv3(i, j, k, repvI, repvJ, repvK, a))
#define repv_5(i, j, k, l, a)                                                  \\
  repss8(i, j, k, l, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]))
#define repv_6(i, j, k, l, m, a)                                               \\
  repss10(i, j, k, l, m, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]),         \\
          sz(a[0][0][0][0]))
#endif

template <typename T> struct has_rbegin_rend {
private:
  template <typename U>
  static auto check(U &&obj)
      -> decltype(std::rbegin(obj), std::rend(obj), std::true_type{});
  static std::false_type check(...);

public:
  static constexpr bool value = decltype(check(std::declval<T>()))::value;
};
template <typename T>
constexpr bool has_rbegin_rend_v = has_rbegin_rend<T>::value;
template <typename Iterator> class Range {
public:
  Range(Iterator &&begin, Iterator &&end) noexcept
      : m_begin(std::forward<Iterator>(begin)),
        m_end(std::forward<Iterator>(end)) {}
  Iterator begin() const noexcept { return m_begin; }
  Iterator end() const noexcept { return m_end; }

private:
  const Iterator m_begin;
  const Iterator m_end;
};
template <typename Iterator>
static inline Range<Iterator> makeRange(Iterator &&begin,
                                        Iterator &&end) noexcept {
  return Range<Iterator>{std::forward<Iterator>(begin),
                         std::forward<Iterator>(end)};
}
template <typename T>
static inline decltype(auto)
makeReversedRange(const std::initializer_list<T> &iniList) noexcept {
  return makeRange(std::rbegin(iniList), std::rend(iniList));
}
template <typename T, typename std::enable_if_t<has_rbegin_rend_v<T>,
                                                std::nullptr_t> = nullptr>
static inline decltype(auto) makeReversedRange(T &&c) noexcept {
  return makeRange(std::rbegin(c), std::rend(c));
} /* rbegin(),
     rend()を持たないものはこっちに分岐させて，エラーメッセージを少なくする*/
template <typename T, typename std::enable_if<!has_rbegin_rend<T>::value,
                                              std::nullptr_t>::type = nullptr>
static inline void makeReversedRange(T &&) noexcept {
  static_assert(has_rbegin_rend<T>::value,
                "Specified argument doesn't have reverse iterator.");
}

// #define use_for

#define form3(k, v, st) for (auto &&[k, v] : st)
#define form4(k, v, st, r)                                                     \\
  for (auto &&[k, v] : range(st.begin(), st.lower_bound(r)))
#define form5(k, v, st, l, r)                                                  \\
  for (auto &&[k, v] : range(st.lower_bound(l), st.lower_bound(r)))
#define form(...) over5(__VA_ARGS__, form5, form4, form3)(__VA_ARGS__)

#define forrm1(st)                                                             \\
  for (auto &&forrm_it = st.rbegin(); forrm_it != st.rend(); ++forrm_it)
#define forrm3(k, v, st)                                                       \\
  for (auto &&forrm_it = st.rbegin(); forrm_it != st.rend(); ++forrm_it)

// 向こう側で
//  ++itか it = st.erase(it)とする
#define fors1(st) for (auto &&it = st.begin(); it != st.end();)
#define fors2(v, st) for (auto &&it = st.begin(); it != st.end();)
#define fors3(v, st, r)                                                        \\
  for (auto &&it = st.begin(); it != st.end() && (*it) < r;)
#define fors4(v, st, l, r)                                                     \\
  for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r;)

#ifdef use_for
#define forslr3(st, a, b)                                                      \\
  for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr4(v, st, a, b)                                                   \\
  for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr5(v, st, r, a, b)                                                \\
  for (auto &&forslr_it = st.begin();                                          \\
       forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)
#define forslr6(v, st, l, r, a, b)                                             \\
  for (auto &&forslr_it = st.lower_bound(l);                                   \\
       forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)

#define fora_f_init_2(a, A) ;
#define fora_f_init_3(fora_f_i, a, A) auto &&a = A[fora_f_i];
#define fora_f_init_4(a, b, A, B)                                              \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];
#define fora_f_init_5(fora_f_i, a, b, A, B)                                    \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];
#define fora_f_init_6(a, b, c, A, B, C)                                        \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];
#define fora_f_init_7(fora_f_i, a, b, c, A, B, C)                              \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];
#define fora_f_init_8(a, b, c, d, A, B, C, D)                                  \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];                                                      \\
  auto &&d = D[fora_f_i];
#define fora_f_init_9(fora_f_i, a, b, c, d, A, B, C, D)                        \\
  auto &&a = A[fora_f_i];                                                      \\
  auto &&b = B[fora_f_i];                                                      \\
  auto &&c = C[fora_f_i];                                                      \\
  auto &&d = D[fora_f_i];
#define fora_f_init(...)                                                       \\
  over9(__VA_ARGS__, fora_f_init_9, fora_f_init_8, fora_f_init_7,              \\
        fora_f_init_6, fora_f_init_5, fora_f_init_4, fora_f_init_3,            \\
        fora_f_init_2)(__VA_ARGS__)

#define forr_init_2(a, A) auto &&a = A[forr_i];
#define forr_init_3(forr_i, a, A) auto &&a = A[forr_i];
#define forr_init_4(a, b, A, B)                                                \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];
#define forr_init_5(forr_i, a, b, A, B)                                        \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];
#define forr_init_6(a, b, c, A, B, C)                                          \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];
#define forr_init_7(forr_i, a, b, c, A, B, C)                                  \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];
#define forr_init_8(a, b, c, d, A, B, C, D)                                    \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];                                                        \\
  auto &&d = D[forr_i];
#define forr_init_9(forr_i, a, b, c, d, A, B, C, D)                            \\
  auto &&a = A[forr_i];                                                        \\
  auto &&b = B[forr_i];                                                        \\
  auto &&c = C[forr_i];                                                        \\
  auto &&d = D[forr_i];
#define forr_init(...)                                                         \\
  over9(__VA_ARGS__, forr_init_9, forr_init_8, forr_init_7, forr_init_6,       \\
        forr_init_5, forr_init_4, forr_init_3, forr_init_2)(__VA_ARGS__)

#define forp_init3(k, v, S)                                                    \\
  auto &&k = S[forp_i].first;                                                  \\
  auto &&v = S[forp_i].second;
#define forp_init4(forp_i, k, v, S)                                            \\
  auto &&k = S[forp_i].first;                                                  \\
  auto &&v = S[forp_i].second;
#define forp_init(...)                                                         \\
  over4(__VA_ARGS__, forp_init4, forp_init3, forp_init2,                       \\
        forp_init1)(__VA_ARGS__)

#define forrm_init(k, v, ...)                                                  \\
  auto &&k = (*forrm_it).fi;                                                   \\
  auto &&v = (*forrm_it).se;
#define fors_init(v, ...) auto &&v = (*it);

#define forlr_init(a, A, ngl, ngr)                                             \\
  auto a = A[forlr_i];                                                         \\
  auto prev = forlr_i ? A[forlr_i - 1] : ngl;                                  \\
  auto next = forlr_i + 1 < rep2lim ? A[forlr_i + 1] : ngr;
#define forslr_init4(a, A, ngl, ngr)                                           \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init5(a, A, r, ngl, ngr)                                        \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init6(a, A, l, r, ngl, ngr)                                     \\
  auto a = (*forslr_it);                                                       \\
  auto prev = (forslr_it != A.begin()) ? (*std::prev(forslr_it)) : ngl;        \\
  auto next = (forslr_it != std::prev(A.end())) ? (*std::next(forslr_it)) : ngr;
#define forslr_init(...)                                                       \\
  over6(__VA_ARGS__, forslr_init6, forslr_init5, forslr_init4)(__VA_ARGS__);

// こうしないとmapがおかしくなる
#define fora_f_2(a, A) for (auto &&a : A)
#define fora_f_3(fora_f_i, a, A) rep(fora_f_i, sz(A))
#define fora_f_4(a, b, A, B) rep(fora_f_i, sz(A))
#define fora_f_5(fora_f_i, a, b, A, B) rep(fora_f_i, sz(A))
#define fora_f_6(a, b, c, A, B, C) rep(fora_f_i, sz(A))
#define fora_f_7(fora_f_i, a, b, c, A, B, C) rep(fora_f_i, sz(A))
#define fora_f_8(a, b, c, d, A, B, C, D) rep(fora_f_i, sz(A))
#define fora_f_9(fora_f_i, a, b, c, d, A, B, C, D) rep(fora_f_i, sz(A))

#define forr_2(a, A) rer(forr_i, sz(A) - 1)
#define forr_3(forr_i, a, A) rer(forr_i, sz(A) - 1)
#define forr_4(a, b, A, B) rer(forr_i, sz(A) - 1)
#define forr_5(forr_i, a, b, A, B) rer(forr_i, sz(A) - 1)
#define forr_6(a, b, c, A, B, C) rer(forr_i, sz(A) - 1)
#define forr_7(forr_i, a, b, c, A, B, C) rer(forr_i, sz(A) - 1)
#define forr_8(a, b, c, d, A, B, C, D) rer(forr_i, sz(A) - 1)
#define forr_9(forr_i, a, b, c, d, A, B, C, D) rer(forr_i, sz(A) - 1)
#endif
// ↑@オーバーロード隔離

// rep系はインデックス、for系は中身
#define rep(...) over4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define rer(...) over4(__VA_ARGS__, rer4, rer3, rer2, )(__VA_ARGS__)

// 自分込みで残りがREM以上の間ループを回す
#define rem(i, N, REM) for (int i = 0; i < N - REM + 1; i++)

// char用のrep
#define repc(i, m, n) for (char i = m, repc3lim = n; i < repc3lim; ++i)
// i,j,k...をnまで見る
#define reps(...) over5(__VA_ARGS__, reps5, reps4, reps3, reps2, )(__VA_ARGS__)
#define repss(...)                                                             \\
  over10(__VA_ARGS__, repss10, a, repss8, a, repss6, a, repss4, a,             \\
         repss2)(__VA_ARGS__)
// vectorのindexを走査する
// repv(i,j,vvi)
#define repv(...)                                                              \\
  over6(__VA_ARGS__, repv_6, repv_5, repv_4, repv_3, repv_2, )(__VA_ARGS__)
#define rerv(i, A) for (int i = sz(A) - 1; i >= 0; i--)
// repvn(dp) nは次元
#define repv1(a) repv(i, a)
#define repv2(a) repv(i, j, a)
#define repv3(a) repv(i, j, k, a)
#define repv4(a) repv(i, j, k, l, a)
#ifdef use_for
#define fora_f(...)                                                            \\
  over9(__VA_ARGS__, fora_f_9, fora_f_8, fora_f_7, fora_f_6, fora_f_5,         \\
        fora_f_4, fora_f_3, fora_f_2)(__VA_ARGS__)
#endif
#define forr(...)                                                              \\
  over9(__VA_ARGS__, forr_9, forr_8, forr_7, forr_6, forr_5, forr_4, forr_3,   \\
        forr_2)(__VA_ARGS__)
// 0~N-2まで見る

#define forar_init(v, rv, A)                                                   \\
  auto &&v = A[forar_i];                                                       \\
  auto &&rv = A[forar_i + 1];
#define forar(v, rv, A) rep(forar_i, sz(A) - 1)

#if __cplusplus >= 201703L
template <size_t M_SZ, bool indexed, class Iterator, class T, class U = T,
          class V = T, class W = T>
class ite_vec_merge : public Iterator {
  std::size_t i = 0;
  vector<T> &A;
  vector<U> &B;
  vector<V> &C;
  vector<W> &D;

public:
  ite_vec_merge(Iterator ita, vector<T> &A)
      : Iterator(ita), A(A), B(A), C(A), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B)
      : Iterator(ita), A(A), B(B), C(A), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C)
      : Iterator(ita), A(A), B(B), C(C), D(A) {}
  ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C,
                vector<W> &D)
      : Iterator(ita), A(A), B(B), C(C), D(D) {}
  auto &operator++() {
    ++i;
    this->Iterator::operator++();
    return *this;
  }
  auto operator*() const noexcept {
    if constexpr (!indexed && M_SZ == 1) {
      return tuple<T &>(A[i]);
    } else if constexpr (!indexed && M_SZ == 2) {
      return tuple<T &, U &>(A[i], B[i]);
    } else if constexpr (!indexed && M_SZ == 3) {
      return tuple<T &, U &, V &>(A[i], B[i], C[i]);
    } else if constexpr (!indexed && M_SZ == 4) {
      return tuple<T &, U &, V &, W &>(A[i], B[i], C[i], D[i]);
    } else if constexpr (indexed && M_SZ == 1) {
      return tuple<int, T &>(i, A[i]);
    } else if constexpr (indexed && M_SZ == 2) {
      return tuple<int, T &, U &>(i, A[i], B[i]);
    } else if constexpr (indexed && M_SZ == 3) {
      return tuple<int, T &, U &, V &>(i, A[i], B[i], C[i]);
    } else if constexpr (indexed && M_SZ == 4) {
      return tuple<int, T &, U &, V &, W &>(i, A[i], B[i], C[i], D[i]);
    } else {
      assert(0);
      return tuple<int>(i);
    }
  }
};
template <size_t M_SZ, bool indexed, class T, class U = T, class V = T,
          class W = T>
class vec_merge {
  vector<T> &a;
  vector<U> &b;
  vector<V> &c;
  vector<W> &d;

public:
  vec_merge(vector<T> &a) : a(a), b(a), c(a), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b) : a(a), b(b), c(a), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b, vector<V> &c)
      : a(a), b(b), c(c), d(a) {}
  vec_merge(vector<T> &a, vector<U> &b, vector<V> &c, vector<W> &d)
      : a(a), b(b), c(c), d(d) {}
  auto begin() const {
    if constexpr (M_SZ == 1) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a};
    } else if constexpr (M_SZ == 2) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b};
    } else if constexpr (M_SZ == 3) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b, c};
    } else if constexpr (M_SZ == 4) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a, b, c, d};
    } else {
      assert(0);
      return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{
          std::begin(a), a};
    }
  }
  auto end() const {
    if constexpr (M_SZ == 1) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a};
    } else if constexpr (M_SZ == 2) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b};
    } else if constexpr (M_SZ == 3) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b, c};
    } else if constexpr (M_SZ == 4) {
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a, b, c, d};
    } else {
      assert(0);
      return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{
          std::end(a), a};
    }
  }
};
#endif
#define fora_2(a, A) for (auto &&a : A)

#if __cplusplus >= 201703L
#define fora_3(i, a, A)                                                        \\
  for (auto [i, a] : vec_merge<1, true, decl_t<decltype(A)>>(A))
#define fora_4(a, b, A, B)                                                     \\
  for (auto [a, b] :                                                           \\
       vec_merge<2, false, decl_t<decltype(A)>, decl_t<decltype(B)>>(A, B))
#define fora_5(i, a, b, A, B)                                                  \\
  for (auto [i, a, b] :                                                        \\
       vec_merge<2, true, decl_t<decltype(A)>, decl_t<decltype(B)>>(A, B))
#define fora_6(a, b, c, A, B, C)                                               \\
  for (auto [a, b, c] :                                                        \\
       vec_merge<3, false, decl_t<decltype(A)>, decl_t<decltype(B)>,           \\
                 decl_t<decltype(C)>>(A, B, C))
#define fora_7(i, a, b, c, A, B, C)                                            \\
  for (auto [i, a, b, c] :                                                     \\
       vec_merge<3, true, decl_t<decltype(A)>, decl_t<decltype(B)>,            \\
                 decl_t<decltype(C)>>(A, B, C))
#define fora_8(a, b, c, d, A, B, C, D)                                         \\
  for (auto [a, b, c, d] :                                                     \\
       vec_merge<4, false, decl_t<decltype(A)>, decl_t<decltype(B)>,           \\
                 decl_t<decltype(C)>, decl_t<decltype(D)>>(A, B, C, D))
#define fora_9(i, a, b, c, d, A, B, C, D)                                      \\
  for (auto [i, a, b, c, d] :                                                  \\
       vec_merge<4, true, decl_t<decltype(A)>, decl_t<decltype(B)>,            \\
                 decl_t<decltype(C)>, decl_t<decltype(D)>>(A, B, C, D))
#endif
// 構造化束縛ver
// 1e5要素で40ms程度
// 遅いときはfora_fを使う
#define fora(...)                                                              \\
  over9(__VA_ARGS__, fora_9, fora_8, fora_7, fora_6, fora_5, fora_4, fora_3,   \\
        fora_2)(__VA_ARGS__)
// #define forr(v, a) for(auto&& v : makeReversedRange(a))
// 参照を取らない
/*@formatter:off*/
#ifdef use_for
template <class U> vector<U> to1d(vector<U> &a) { return a; }
template <class U> auto to1d(vector<vector<U>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      res.push_back(a2);
  return res;
}
template <class U> vector<U> to1d(vector<vector<vector<U>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        res.push_back(a3);
  return res;
}
template <class U> vector<U> to1d(vector<vector<vector<vector<U>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          res.push_back(a4);
  return res;
}
template <class U>
vector<U> to1d(vector<vector<vector<vector<vector<U>>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          for (auto &&a5 : a4)
            res.push_back(a5);
  return res;
}
template <class U>
vector<U> to1d(vector<vector<vector<vector<vector<vector<U>>>>>> &a) {
  vector<U> res;
  for (auto &&a1 : a)
    for (auto &&a2 : a1)
      for (auto &&a3 : a2)
        for (auto &&a4 : a3)
          for (auto &&a5 : a4)
            for (auto &&a6 : a5)
              res.push_back(a6);
  return res;
}
#define forv(a, b) for (auto a : to1d(b))
// インデックスを前後含めて走査
#define ring(i, s, len)                                                        \\
  for (int i = s, prev = (s == 0) ? len - 1 : s - 1,                           \\
           next = (s == len - 1) ? 0 : s + 1, cou = 0;                         \\
       cou < len;                                                              \\
       cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
// 値と前後を見る
#define ringv(v, d)                                                            \\
  index_ = 0;                                                                  \\
  for (auto prev = d[sz(d) - 1], next = (int)d.size() > 1 ? d[1] : d[0],       \\
            v = d[0];                                                          \\
       index_ < sz(d); index_++, prev = v, v = next,                           \\
            next = (index_ >= sz(d) - 1 ? d[0] : d[index_ + 1]))
// 左右をnext prevで見る 0の左と nの右
#define forlr(v, d, banpei_l, banpei_r) rep(forlr_i, sz(d))
#endif

#define forrm(...)                                                             \\
  over5(__VA_ARGS__, forrm5, forrm4, forrm3, forrm2, forrm1)(__VA_ARGS__)
#define fors(...) over4(__VA_ARGS__, fors4, fors3, fors2, fors1)(__VA_ARGS__)
#define forslr(...)                                                            \\
  over6(__VA_ARGS__, forslr6, forslr5, forslr4, forslr3)(__VA_ARGS__)
#define forp3(k, v, st) rep(forp_i, sz(st))
#define forp4(forp_i, k, v, st) rep(forp_i, sz(st))
#define forp(...) over4(__VA_ARGS__, forp4, forp3)(__VA_ARGS__)

// to_vec(rep(i, N))のように使い
// iが走査した値を持つvectorを返す
#define to_vec2(type, my_for)                                                  \\
  [&]() {                                                                      \\
    vector<type> ret;                                                          \\
    my_for { ret.push_back(i); }                                               \\
    return ret;                                                                \\
  }()
#define to_vec1(my_for) to_vec2(int, my_for)
#define to_vec(...) over2(__VA_ARGS__, to_vec2, to_vec1)(__VA_ARGS__)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010

const double PI = 3.1415926535897932384626433832795029L;

constexpr bool ev(ll a) { return !(a & 1); }
constexpr bool od(ll a) { return (a & 1); }
//@拡張系 こう出来るべきというもの

// 埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
template <> class hash<std::pair<signed, signed>> {
public:
  size_t operator()(const std::pair<signed, signed> &x) const {
    return hash<ll>()(((ll)x.first << 32) | x.second);
  }
};
template <> class hash<std::pair<ll, ll>> {
public
    : /*大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断*/
  size_t operator()(const std::pair<ll, ll> &x) const {
    return hash<ll>()(((ll)x.first << 32) | x.second);
  }
};
} // namespace std
// stream まとめ
/*@formatter:off*/
istream &operator>>(istream &iss, P &a) {
  iss >> a.first >> a.second;
  return iss;
}
template <typename T> istream &operator>>(istream &iss, vector<T> &vec_) {
  for (T &x : vec_)
    iss >> x;
  return iss;
}
template <class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
  os << p.fi << " " << p.se;
  return os;
}
ostream &operator<<(ostream &os, T p) {
  os << get<0>(p) << " " << get<1>(p) << " " << get<2>(p);
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << get<0>(p) << " " << get<1>(p) << " " << get<2>(p) << " " << get<3>(p);
  return os;
}
template <typename T> ostream &operator<<(ostream &os, const vector<T> &vec_) {
  for (ll i = 0; i < vec_.size(); ++i)
    os << vec_[i] << (i + 1 == vec_.size() ? "" : " ");
  return os;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<vector<T>> &vec_) {
  for (ll i = 0; i < vec_.size(); ++i) {
    for (ll j = 0; j < vec_[i].size(); ++j) {
      os << vec_[i][j] << " ";
    }
    os << endl;
  }
  return os;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const unordered_map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}

template <typename T, typename U> string deb_tos(const unordered_map<T, U> &m) {
  map<T, U> a;
  for (auto &&v : m) {
    a[v.first] = v.second;
  }
  stringstream ss;
  ss << a;
  return ss.str();
}
template <class T> ostream &operator<<(ostream &os, const set<T> &s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, const mset<T> &s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, const deque<T> &a) {
  fora(v, a) { os << v << " "; }
  return os;
}
ostream &operator<<(ostream &os, const vector<vector<char>> &vec_) {
  rep(h, sz(vec_)) {
    rep(w, sz(vec_[0])) { os << vec_[h][w]; }
    os << endl;
  }
  return os;
}
ostream &operator<<(ostream &os,
                    const vector<vector<my_optional<char>>> &vec_) {
  rep(h, sz(vec_)) {
    rep(w, sz(vec_[0])) { os << vec_[h][w]; }
    os << endl;
  }
  return os;
}
// tuple "," 区切り
#if __cplusplus >= 201703L
template <class Tuple, size_t... Indicies>
void helper_os_tuple(ostream &os, const Tuple &tup,
                     index_sequence<Indicies...>) {
  stringstream ss;
  auto f = [&](auto a) { ss << ", " << a; };
  (..., f(get<Indicies>(tup)));
  os << ss.str().substr(2);
}
template <class... T> ostream &operator<<(ostream &os, const tuple<T...> &tup) {
  helper_os_tuple(os, tup, index_sequence_for<T...>{});
  return os;
}
#endif

template <class T> struct range_now {
  int l;
  vector<T> A;
  range_now(vector<T> &&A, int l) : A(A), l(l) {}
};
/*@formatter:off*/
// template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>&
// a) {fora_f(v,a)os<<v<<endl;return os;}
template <typename W, typename H> void resize(W &vec_, const H head) {
  vec_.resize(head);
}
template <typename W, typename H, typename... T>
void resize(W &vec_, const H &head, const T... tail) {
  vec_.resize(head);
  for (auto &v : vec_)
    resize(v, tail...);
}
// #define use_for_each  //_each _all_of _any_of _none_of _find_if _rfind_if
// _contains _count_if _erase_if _entry_if

#ifdef use_for_each
// todo Atcoderの過去問がc++17に対応したら
// for_each以外はconst & (呼び出し側のラムダも)
template <typename T, typename F> bool all_of2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : v) {
      if (!all_of2(v_, f))
        return false;
    }
    return true;
  } else {
    return f(v);
  }
}
template <typename T, typename F> bool any_of2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : v) {
      if (any_of2(v_, f))
        return true;
    }
    return false;
  } else {
    return f(v);
  }
}
template <typename T, typename F> bool none_of2(const T &v, F f) {
  return all_of2(v, [&](auto a) { return !f(a); });
}

// 存在しない場合
// 1次元 Nを返す
// 多次元(-1,-1,..)を返す
template <typename T, typename F, require_t(has_value_type<T>::value)>
ll find_if2(const T &v, F f) {
  rep(i, sz(v)) {
    if (f(v[i]))
      return i;
  }
  return sz(v);
}
template <typename T, typename F>
tuple<int, int> find_if2(const vector<vector<T>> &v, F f) {
  rep(i, sz(v)) {
    rep(j, sz(v[i])) {
      if (f(v[i][j])) {
        return tuple<int, int>(i, j);
      }
    }
  }
  return tuple<int, int>(-1, -1);
}
template <typename T, typename F>
auto find_if2(const vector<vector<vector<T>>> &v, F f) {
  rep(i, sz(v)) {
    if (auto ret = find_if2(v[i], f); get<0>(ret) != -1) {
      return tuple_cat(tuple<int>(i), ret);
    }
  }
  auto bad = tuple_cat(tuple<int>(-1), find_if2(v[0], f));
  return bad;
}
template <class T, class F> auto find_if2(const range_now<T> &v, F f) {
  return find_if2(v.A, f) + v.l;
}
// 存在しない場合
// 1次元 -1を返す
// 多次元-1を返す
template <typename T, typename F, require_t(has_value_type<T>::value)>
ll rfind_if2(const T &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return i;
  }
  return -1;
}
template <typename T, typename F>
tuple<int, int> rfind_if2(const vector<vector<T>> &v, F f) {
  rer(i, sz(v) - 1) {
    rer(j, sz(v[i]) - 1) {
      if (f(v[i][j])) {
        return tuple<int, int>(i, j);
      }
    }
  }
  return tuple<int, int>(-1, -1);
}
template <typename T, typename F>
auto rfind_if2(const vector<vector<vector<T>>> &v, F f) {
  rer(i, sz(v) - 1) {
    if (auto ret = rfind_if2(v[i], f); get<0>(ret) != -1) {
      return tuple_cat(tuple<int>(i), ret);
    }
  }
  auto bad = tuple_cat(tuple<int>(-1), rfind_if2(v[0], f));
  return bad;
}

// todo まとめられそう string,vector全般
template <class T> bool contains(const string &s, const T &v) {
  return s.find(v) != string::npos;
}
template <typename T> bool contains(const vector<T> &v, const T &val) {
  return std::find(v.begin(), v.end(), val) != v.end();
}
template <typename T, typename F> bool contains_if2(const vector<T> &v, F f) {
  return find_if(v.begin(), v.end(), f) != v.end();
}
template <typename T, typename F> ll count_if2(const T &v, F f) {
  if constexpr (has_value_type<T>::value) {
    ll ret = 0;
    for (auto &&v_ : v) {
      ret += count_if2(v_, f);
    }
    return ret;
  } else {
    return f(v);
  }
}
template <typename T, typename F> void for_each2(T &a, F f) {
  if constexpr (has_value_type<T>::value) {
    for (auto &&v_ : a)
      for_each2(v_, f);
  } else {
    f(a);
  }
}

template <typename W> ll count_od(const vector<W> &a) {
  return count_if2(a, [](ll v) { return v & 1; });
}
template <typename W> ll count_ev(const vector<W> &a) {
  return count_if2(a, [](ll v) { return !(v & 1); });
}
// 削除した後のvectorを返す
template <typename T, typename F> vector<T> erase_if2(const vector<T> &v, F f) {
  vector<T> nv;
  rep(i, sz(v)) {
    if (!f(v[i])) {
      nv.push_back(v[i]);
    }
  }
  return nv;
}
template <typename T, typename F>
vector<vector<T>> erase_if2(const vector<vector<T>> &v, F f) {
  vector<vector<T>> res;
  rep(i, sz(v)) { res[i] = erase_if2(v[i], f); }
  return res;
}

template <typename T, typename F> vector<T> entry_if2(const vector<T> &v, F f) {
  vector<T> nv;
  rep(i, sz(v)) {
    if (f(v[i])) {
      nv.push_back(v[i]);
    }
  }
  return nv;
}
template <typename T, typename F>
vector<vector<T>> entry_if2(const vector<vector<T>> &v, F f) {
  vector<vector<T>> res;
  rep(i, sz(v)) { res[i] = entry_if2(v[i], f); }
  return res;
}

template <typename T, typename F> ll l_rfind_if(const vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return i;
  }
  return -1;
}
template <typename T, typename F> bool l_contains_if(const vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (f(v[i]))
      return true;
  }
  return false;
}
template <class A, class B, class C> auto t_all_of(A a, B b, C c) {
  return std::all_of(a, b, c);
}
template <class A, class B, class C> auto t_any_of(A a, B b, C c) {
  return std::any_of(a, b, c);
}
template <class A, class B, class C> auto t_none_of(A a, B b, C c) {
  return std::none_of(a, b, c);
}
template <class A, class B, class C> auto t_find_if(A a, B b, C c) {
  return std::find_if(a, b, c);
}
template <class A, class B, class C> auto t_count_if(A a, B b, C c) {
  return std::count_if(a, b, c);
}

#define all_of_s__2(a, right) (t_all_of(ALL(a), lamr(right)))
#define all_of_s__3(a, v, siki) (t_all_of(ALL(a), [&](auto v) { return siki; }))
#define all_of_s(...) over3(__VA_ARGS__, all_of_s__3, all_of_s__2)(__VA_ARGS__)

// all_of(A, %2);
// all_of(A, a, a%2);
#define all_of__2(a, right) all_of2(a, lamr(right))
#define all_of__3(a, v, siki) all_of2(a, [&](auto v) { return siki; })
#define all_of(...) over3(__VA_ARGS__, all_of__3, all_of__2)(__VA_ARGS__)
#define all_of_f(a, f) all_of2(a, f)

#define any_of_s__2(a, right) (t_any_of(ALL(a), lamr(right)))
#define any_of_s__3(a, v, siki) (t_any_of(ALL(a), [&](auto v) { return siki; }))
#define any_of_s(...) over3(__VA_ARGS__, any_of_s__3, any_of_s__2)(__VA_ARGS__)

#define any_of__2(a, right) any_of2(a, lamr(right))
#define any_of__3(a, v, siki) any_of2(a, [&](auto v) { return siki; })
#define any_of(...) over3(__VA_ARGS__, any_of__3, any_of__2)(__VA_ARGS__)
#define any_of_f(a, f) any_of2(a, f)

#define none_of_s__2(a, right) (t_none_of(ALL(a), lamr(right)))
#define none_of_s__3(a, v, siki)                                               \\
  (t_none_of(ALL(a), [&](auto v) { return siki; }))
#define none_of_s(...)                                                         \\
  over3(__VA_ARGS__, none_of_s__3, none_of_s__2)(__VA_ARGS__)

#define none_of__2(a, right) none_of2(a, lamr(right))
#define none_of__3(a, v, siki) none_of2(a, [&](auto v) { return siki; })
#define none_of(...) over3(__VA_ARGS__, none_of__3, none_of__2)(__VA_ARGS__)
#define none_of_f(a, f) none_of2(a, f)

#define find_if_s__2(a, right) (t_find_if(ALL(a), lamr(right)) - a.begin())
#define find_if_s__3(a, v, siki)                                               \\
  (t_find_if(ALL(a), [&](auto v) { return siki; }) - a.begin())
#define find_if_s(...)                                                         \\
  over3(__VA_ARGS__, find_if_s__3, find_if_s__2)(__VA_ARGS__)

#define find_if__2(a, right) find_if2(a, lamr(right))
#define find_if__3(a, v, siki) find_if2(a, [&](auto v) { return siki; })
#define find_if__4(a, l, v, siki)                                              \\
  (find_if2(decltype(a)(a.begin() + l, a.end()),                               \\
            [&](auto v) { return siki; }) +                                    \\
   l)
#define find_if(...)                                                           \\
  over4(__VA_ARGS__, find_if__4, find_if__3, find_if__2)(__VA_ARGS__)
#define find_if_f(a, f) find_if2(a, f)

#define rfind_if_s__2(a, right) l_rfind_if(a, lamr(right))
#define rfind_if_s__3(a, v, siki) l_rfind_if(a, [&](auto v) { return siki; })
#define rfind_if_s(...)                                                        \\
  over3(__VA_ARGS__, rfind_if_s__3, rfind_if_s__2)(__VA_ARGS__)

#define rfind_if__2(a, right) rfind_if2(a, lamr(right))
#define rfind_if__3(a, v, siki) rfind_if2(a, [&](auto v) { return siki; })
#define rfind_if(...) over3(__VA_ARGS__, rfind_if__3, rfind_if__2)(__VA_ARGS__)
#define rfind_if_f(a, f) rfind_if2(a, f)

#define contains_if_s__2(a, right) l_contains_if(a, lamr(right))
#define contains_if_s__3(a, v, siki)                                           \\
  l_contains_if(a, [&](auto v) { return siki; })
#define contains_if_s(...)                                                     \\
  over3(__VA_ARGS__, contains_if_s__3, contains_if_s__2)(__VA_ARGS__)

#define contains_if__2(a, right) contains_if2(a, lamr(right))
#define contains_if__3(a, v, siki) contains_if2(a, [&](auto v) { return siki; })
#define contains_if(...)                                                       \\
  over3(__VA_ARGS__, contains_if__3, contains_if__2)(__VA_ARGS__)
#define contains_if_f(a, f) contains_if2(a, f)

#define count_if_s__2(a, right) (t_count_if(ALL(a), lamr(right)))
#define count_if_s__3(a, v, siki)                                              \\
  (t_count_if(ALL(a), [&](auto v) { return siki; }))
#define count_if_s(...)                                                        \\
  over3(__VA_ARGS__, count_if_s__3, count_if_s__2)(__VA_ARGS__)

#define count_if__2(a, right) count_if2(a, lamr(right))
#define count_if__3(a, v, siki) count_if2(a, [&](auto v) { return siki; })
#define count_if(...) over3(__VA_ARGS__, count_if__3, count_if__2)(__VA_ARGS__)
#define count_if_f(a, f) count_if2(a, f)

// vector<vi>で、viに対して操作
#define for_each_s__2(a, right)                                                \\
  do {                                                                         \\
    fora(v, a) { v right; }                                                    \\
  } while (0)
#define for_each_s__3(a, v, shori)                                             \\
  do {                                                                         \\
    fora(v, a) { shori; }                                                      \\
  } while (0)
#define for_each_s(...)                                                        \\
  over3(__VA_ARGS__, for_each_s__3, for_each_s__2)(__VA_ARGS__)

// vector<vi>で、intに対して操作
#define for_each__2(a, right) for_each2(a, lamr(right))
#define for_each__3(a, v, shori) for_each2(a, [&](auto &v) { shori; })
#define for_each(...) over3(__VA_ARGS__, for_each__3, for_each__2)(__VA_ARGS__)
#define for_each_f(a, f) for_each2(a, f);

template <class T, class F> vector<T> help_for_eached(const vector<T> &A, F f) {
  vector<T> ret = A;
  for_each(ret, v, f(v));
  return ret;
}
#define for_eached__2(a, right) help_for_eached(a, lamr(right))
#define for_eached__3(a, v, shori) help_for_eached(a, lam(v, shori))
#define for_eached(...)                                                        \\
  over3(__VA_ARGS__, for_eached__3, for_eached__2)(__VA_ARGS__)
#define for_eached_f(a, f) for_eached2(a, f);

#define each for_each
#define eached for_eached

// #define erase_if_s__2(a, right) l_erase_if2(a,lamr(right))
// #define erase_if_s__3(a, v, siki) l_erase_if2(a,[&](auto v){return siki;})
// #define erase_if_s(...)
// over3(__VA_ARGS__,erase_if_s__3,erase_if_s__2)(__VA_ARGS__)

#define erase_if__2(a, right) erase_if2(a, lamr(right))
#define erase_if__3(a, v, siki) erase_if2(a, [&](auto v) { return siki; })
#define erase_if(...) over3(__VA_ARGS__, erase_if__3, erase_if__2)(__VA_ARGS__)
#define erase_if_f(a, f) erase_if2(a, f)

// #define entry_if_s__2(a, right) l_entry_if2(a,lamr(right))
// #define entry_if_s__3(a, v, siki) l_entry_if2(a,[&](auto v){return siki;})
// #define entry_if_s(...)
// over3(__VA_ARGS__,entry_if_s__3,entry_if_s__2)(__VA_ARGS__)

#define entry_if__2(a, right) entry_if2(a, lamr(right))
#define entry_if__3(a, v, siki) entry_if2(a, [&](auto v) { return siki; })
#define entry_if(...) over3(__VA_ARGS__, entry_if__3, entry_if__2)(__VA_ARGS__)
#define entry_if_f(a, f) entry_if2(a, f)

#endif

// 機能削除 todo
// string.replace()は長さを指定するため、間違えやすい

/*@formatter:off*/
void replace(str &a, char key, char v) { replace(ALL(a), key, v); }
template <class T, class U> void my_replace(T &s, int l, int r, const U &t) {
  s.replace(l, r - l, t);
}
#define replace my_replace
template <class T, class U, class W> void replace(vector<W> &a, T key, U v) {
  rep(i, sz(a)) if (a[i] == key) a[i] = v;
}
template <class T, class U, class W>
void replace(vector<vector<W>> &A, T key, U v) {
  rep(i, sz(A)) replace(A[i], key, v);
}
void replace(str &a, char key, str v) {
  if (v == "")
    a.erase(remove(ALL(a), key), a.end());
}
// keyと同じかどうか01で置き換える
template <class T, class U> void replace(vector<T> &a, U k) {
  rep(i, sz(a)) a[i] = a[i] == k;
}
template <class T, class U> void replace(vector<vector<T>> &a, U k) {
  rep(i, sz(a)) rep(j, sz(a[0])) a[i][j] = a[i][j] == k;
}
void replace(str &a) {
  int dec = 0;
  if ('a' <= a[0] && a[0] <= 'z')
    dec = 'a';
  if ('A' <= a[0] && a[0] <= 'Z')
    dec = 'A';
  fora(v, a) { v -= dec; }
}
void replace(string &a, char key) {
  string res;
  for (auto c : a) {
    if (c != key)
      res.push_back(c);
  }
  swap(a, res);
}
void replace(str &a, str key, str v) {
  stringstream t;
  ll kn = sz(key);
  std::string::size_type Pos(a.find(key));
  ll l = 0;
  while (Pos != std::string::npos) {
    t << a.substr(l, Pos - l);
    t << v;
    l = Pos + kn;
    Pos = a.find(key, Pos + kn);
  }
  t << a.substr(l, sz(a) - l);
  a = t.str();
}

template <class T> bool is_permutation(vector<T> &a, vector<T> &b) {
  return is_permutation(ALL(a), ALL(b));
}
template <class T> bool next_permutation(vector<T> &a) {
  return next_permutation(ALL(a));
}
vi iota(ll s, ll len) {
  vi ve(len);
  iota(ALL(ve), s);
  return ve;
}

//[iterator, iterator)等と渡す
// vectorに変換もできる
#if __cplusplus >= 201703L
template <class I, class J> struct body_range {
  I itl;
  J itr;
  body_range(I it, J end) : itl(it), itr(end) {}
  I begin() { return itl; }
  I end() { return itr; }
  // 毎回コピーする
  operator vector<typename I::value_type>() {
    return vector<typename I::value_type>(itl, itr);
  }
};
template <class I, class J, require_t(is_integral_v<I>)>
vector<int> range(I l, J r) {
  return iota(l, r - l);
}
template <class I, require_t(is_integral_v<I>)> vector<int> range(I r) {
  return iota(0, r);
}
template <class I, require_t(!is_integral_v<I>)> auto range(I itl, I itr) {
  return body_range(itl, itr);
}
#endif
/*@formatter:off*/

// tuple<vecs...> = take_vt(vt)
#if __cplusplus >= 201703L
template <class Tup_l, class Tup_r, size_t... Indicies>
auto helper_take_vector_tuple(Tup_l &res, const Tup_r &rhs,
                              index_sequence<Indicies...>) {
  (..., get<Indicies>(res).emplace_back(get<Indicies>(rhs)));
}
template <class... T> auto take_vector_tuple(const vector<tuple<T...>> &elems) {
  tuple<vector<T>...> res;
  for (auto &&elem : elems) {
    helper_take_vector_tuple(res, elem, index_sequence_for<T...>{});
  }
  return res;
}
// tie_vt(A,B,C) = vt;
template <class... T> struct tie_vector_tuple {
  tuple<vector<T> &...> v;
  tie_vector_tuple(vector<T> &...args) : v(args...) {}
  template <class Tup, size_t... Indicies>
  void helper_emplace(int i, Tup &rhs, index_sequence<Indicies...>) {
    (..., (get<Indicies>(v).emplace_back(get<Indicies>(rhs))));
  }
  template <size_t... Indicies> void clear(index_sequence<Indicies...>) {
    (..., (get<Indicies>(v).clear()));
  }
  void operator=(vector<tuple<T...>> &rhs) {
    clear(index_sequence_for<T...>{});
    rep(i, sz(rhs)) { helper_emplace(i, rhs[i], index_sequence_for<T...>{}); }
  }
};
/*@formatter:off*/
// vt = make_vt(A, B, C)
template <class H, class... T>
auto make_vector_tuple(const vector<H> &head, const vector<T> &...tails) {
  vector<tuple<H, T...>> res;
  rep(i, sz(head)) { res.emplace_back(head[i], tails[i]...); }
  return res;
}
#endif
/*@formatter:off*/
#define mvt make_vector_tuple

#define tie_vt tie_vector_tuple
#define make_vt make_vector_tuple
#define take_vt take_vector_tuple

/*@formatter:off*/
// 基本ソート@0
template <class T> void sort(vector<T> &a, int l = -1, int r = -1) {
  set_lr12(l, r, sz(a));
  fast_sort(a.begin() + l, a.begin() + r);
}
template <class T> void rsort(vector<T> &a, int l = -1, int r = -1) {
  set_lr12(l, r, sz(a));
  fast_sort(a.begin() + l, a.begin() + r, greater<T>());
};
// f(tup)の大きさでソート
template <class U, class F> void sort(vector<U> &a, F f) {
  fast_sort(ALL(a), [&](U l, U r) { return f(l) < f(r); });
};
template <class U, class F> void rsort(vector<U> &a, F f) {
  fast_sort(ALL(a), [&](U l, U r) { return f(l) > f(r); });
};

enum tuple_comparator { /*tcom_less = 1728,    tcom_greater = 1729, */ fisi = 0,
                        fisd,
                        fdsi,
                        fdsd,
                        sifi,
                        sifd,
                        sdfi,
                        sdfd,
                        fisiti,
                        fisitd,
                        fisdti,
                        fisdtd,
                        fdsiti,
                        fdsitd,
                        fdsdti,
                        fdsdtd,
                        fitisi,
                        fitisd,
                        fitdsi,
                        fitdsd,
                        fdtisi,
                        fdtisd,
                        fdtdsi,
                        fdtdsd,
                        sifiti,
                        sifitd,
                        sifdti,
                        sifdtd,
                        sdfiti,
                        sdfitd,
                        sdfdti,
                        sdfdtd,
                        sitifi,
                        sitifd,
                        sitdfi,
                        sitdfd,
                        sdtifi,
                        sdtifd,
                        sdtdfi,
                        sdfdfd,
                        tifisi,
                        tifisd,
                        tifdsi,
                        tifdsd,
                        tdfisi,
                        tdfisd,
                        tdfdsi,
                        tdfdsd,
                        tisifi,
                        tisifd,
                        tisdfi,
                        tisdfd,
                        tdsifi,
                        tdsifd,
                        tdsdfi,
                        tdsdfd };

// todo 短くする
#define set_4(i0, i1, o0, o1)                                                  \\
  [&](U l, U r) {                                                              \\
    return get<i0>(l) != get<i0>(r) ? get<i0>(l) o0 get<i0>(r)                 \\
                                    : get<i1>(l) o1 get<i1>(r);                \\
  }
#define set_6(i0, i1, i2, o0, o1, o2)                                          \\
  [&](U l, U r) {                                                              \\
    return get<i0>(l) != get<i0>(r)   ? get<i0>(l) o0 get<i0>(r)               \\
           : get<i1>(l) != get<i1>(r) ? get<i1>(l) o1 get<i1>(r)               \\
                                      : get<i2>(l) o2 get<i2>(r);              \\
  }

// functionを返すため少し遅い todo
#if __cplusplus >= 201703L
template <class... T, class U = tuple<T...>>
function<bool(U, U)> get_function2(const vector<tuple<T...>> &a,
                                   tuple_comparator Compare) {
  if constexpr (sizeof...(T) >= 2) {
    int type = Compare;
    if (type == 0)
      return set_4(0, 1, <, <);
    if (type == 1)
      return set_4(0, 1, <, >);
    if (type == 2)
      return set_4(0, 1, >, <);
    if (type == 3)
      return set_4(0, 1, >, >);
    if (type == 4)
      return set_4(1, 0, <, <);
    if (type == 5)
      return set_4(1, 0, <, >);
    if (type == 6)
      return set_4(1, 0, >, <);
    if (type == 7)
      return set_4(1, 0, >, >);
  }
  if constexpr (sizeof...(T) >= 3) {
    int type = Compare - 8;
    if (type == 0)
      return set_6(0, 1, 2, <, <, <);
    else if (type == 1)
      return set_6(0, 1, 2, <, <, >);
    else if (type == 2)
      return set_6(0, 1, 2, <, >, <);
    else if (type == 3)
      return set_6(0, 1, 2, <, >, >);
    else if (type == 4)
      return set_6(0, 1, 2, >, <, <);
    else if (type == 5)
      return set_6(0, 1, 2, >, <, >);
    else if (type == 6)
      return set_6(0, 1, 2, >, >, <);
    else if (type == 7)
      return set_6(0, 1, 2, >, >, >);
    else if (type == 8)
      return set_6(0, 2, 1, <, <, <);
    else if (type == 9)
      return set_6(0, 2, 1, <, <, >);
    else if (type == 10)
      return set_6(0, 2, 1, <, >, <);
    else if (type == 11)
      return set_6(0, 2, 1, <, >, >);
    else if (type == 12)
      return set_6(0, 2, 1, >, <, <);
    else if (type == 13)
      return set_6(0, 2, 1, >, <, >);
    else if (type == 14)
      return set_6(0, 2, 1, >, >, <);
    else if (type == 15)
      return set_6(0, 2, 1, >, >, >);
    else if (type == 16)
      return set_6(1, 0, 2, <, <, <);
    else if (type == 17)
      return set_6(1, 0, 2, <, <, >);
    else if (type == 18)
      return set_6(1, 0, 2, <, >, <);
    else if (type == 19)
      return set_6(1, 0, 2, <, >, >);
    else if (type == 20)
      return set_6(1, 0, 2, >, <, <);
    else if (type == 21)
      return set_6(1, 0, 2, >, <, >);
    else if (type == 22)
      return set_6(1, 0, 2, >, >, <);
    else if (type == 23)
      return set_6(1, 0, 2, >, >, >);
    else if (type == 24)
      return set_6(1, 2, 0, <, <, <);
    else if (type == 25)
      return set_6(1, 2, 0, <, <, >);
    else if (type == 26)
      return set_6(1, 2, 0, <, >, <);
    else if (type == 27)
      return set_6(1, 2, 0, <, >, >);
    else if (type == 28)
      return set_6(1, 2, 0, >, <, <);
    else if (type == 29)
      return set_6(1, 2, 0, >, <, >);
    else if (type == 30)
      return set_6(1, 2, 0, >, >, <);
    else if (type == 31)
      return set_6(1, 2, 0, >, >, >);
    else if (type == 32)
      return set_6(2, 0, 1, <, <, <);
    else if (type == 33)
      return set_6(2, 0, 1, <, <, >);
    else if (type == 34)
      return set_6(2, 0, 1, <, >, <);
    else if (type == 35)
      return set_6(2, 0, 1, <, >, >);
    else if (type == 36)
      return set_6(2, 0, 1, >, <, <);
    else if (type == 37)
      return set_6(2, 0, 1, >, <, >);
    else if (type == 38)
      return set_6(2, 0, 1, >, >, <);
    else if (type == 39)
      return set_6(2, 0, 1, >, >, >);
    else if (type == 40)
      return set_6(2, 1, 0, <, <, <);
    else if (type == 41)
      return set_6(2, 1, 0, <, <, >);
    else if (type == 42)
      return set_6(2, 1, 0, <, >, <);
    else if (type == 43)
      return set_6(2, 1, 0, <, >, >);
    else if (type == 44)
      return set_6(2, 1, 0, >, <, <);
    else if (type == 45)
      return set_6(2, 1, 0, >, <, >);
    else if (type == 46)
      return set_6(2, 1, 0, >, >, <);
    else if (type == 47)
      return set_6(2, 1, 0, >, >, >);
  }
  return [&](U l, U r) { return true; };
}
template <class... T>
void sort(vector<tuple<T...>> &a, tuple_comparator Compare) {
  auto f = get_function2(a, Compare);
  fast_sort(ALL(a), f);
}
#endif
/*@formatter:off*/
#undef set_4
#undef set_6

#if __cplusplus >= 201703L
// sort_tuple
template <class H1, class H2, class... T>
void sortt(vector<H1> &head1, vector<H2> &head2, vector<T> &...tails) {
  auto vec_t = make_vt(head1, head2, tails...);
  sort(vec_t);
  tie_vt(head1, head2, tails...) = vec_t;
}
template <class H1, class H2, class... T>
void rsortt(vector<H1> &head1, vector<H2> &head2, vector<T> &...tails) {
  auto vec_t = make_vt(head1, head2, tails...);
  rsort(vec_t);
  tie_vt(head1, head2, tails...) = vec_t;
}
template <class F, class... T> void helper_sortt_f(F f, vector<T> &...a) {
  auto z = make_vt(a...);
  sort(z, f);
  tie_vt(a...) = z;
}
template <class F, class... T> void helper_rsortt_f(F f, vector<T> &...a) {
  auto z = make_vt(a...);
  rsort(z, f);
  tie_vt(a...) = z;
}
// sortt(a,b,c,d, f)まで対応->
template <class A, class B, class F, require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, F f) {
  helper_sortt_f(f, a, b);
}
template <class A, class B, class F, require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, F f) {
  helper_rsortt_f(f, a, b);
}
template <class A, class B, class C, class F, require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  helper_sortt_f(f, a, b, c);
}
template <class A, class B, class C, class F, require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  helper_rsortt_f(f, a, b, c);
}
template <class A, class B, class C, class D, class F,
          require_t(!is_vector<F>::value)>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d, F f) {
  helper_sortt_f(f, a, b, c, d);
}
template <class A, class B, class C, class D, class F,
          require_t(!is_vector<F>::value)>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d, F f) {
  helper_rsortt_f(f, a, b, c, d);
} //  sort_tuple f }
#endif

/*@formatter:off*/
// 任意引数は取らない
template <class T> vi sorti(const vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(b));
  sortt(b, ind);
  return ind;
}
template <class T> vi rsorti(const vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(b));
  rsortt(b, ind);
  return ind;
}

#if __cplusplus >= 201703L
template <class T, class F> vi sorti(const vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  if constexpr (is_same_v<F, tuple_comparator>) {
    auto f2 = get_function2(a, f);
    auto g = [&](int i, int j) { return f2(a[i], a[j]); };
    fast_sort(ALL(ind), g);
  } else {
    auto g = [&](int i) { return f(a[i]); };
    sort(ind, g);
  }
  return ind;
}
template <class T, class F, require_t(is_function_v<F>)>
vi rsorti(const vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  auto g = [&](int i) { return f(a[i]); };
  rsort(ind, g);
  return ind;
}

// 任意引数にfを渡したい場合は (make_vt(a, b, c), f)
template <class H, class... Ts> vi sortti(vector<H> head, vector<Ts>... tails) {
  vi ind = iota(0, sz(head));
  auto vec = make_vt(head, tails..., ind);
  sort(vec);
  auto rets = take_vt(vec);
  return get<sizeof...(tails) + 1>(rets);
}
template <class H, class... Ts>
vi rsortti(vector<H> head, vector<Ts>... tails) {
  vi ind = iota(0, sz(head));
  auto vec = make_vt(head, tails..., ind);
  rsort(vec);
  auto rets = take_vt(vec);
  return get<sizeof...(tails) + 1>(rets);
}
#endif
/*@formatter:off*/
//@5
void sort(string &a) { sort(ALL(a)); }
void rsort(string &a) { sort(RALL(a)); }
void sort(int &a, int &b) {
  if (a > b)
    swap(a, b);
}
void sort(int &a, int &b, int &c) {
  sort(a, b);
  sort(a, c);
  sort(b, c);
}
void rsort(int &a, int &b) {
  if (a < b)
    swap(a, b);
}
void rsort(int &a, int &b, int &c) {
  rsort(a, b);
  rsort(a, c);
  rsort(b, c);
}
template <class... T, class U> auto sorted(U head, T... a) {
  sort(head, a...);
  return head;
}
template <class... T, class U> auto rsorted(U head, T... a) {
  rsort(head, a...);
  return head;
}

template <class T> vector<T> merge(const vector<T> &a, const vector<T> &b) {
  vector<T> res;
  std::merge(ALL(a), ALL(b), back_inserter(res));
  return res;
}

// firstの値が同じ場合secondが最小の物を残す
template <class T, class U> void unique_min(vector<pair<T, U>> &a) {
  vector<pair<T, U>> res;
  if (sz(a))
    res.push_back(a[0]);
  rep(i, 1, sz(a)) {
    if (res.back().fi == a[i].fi) {
      chmi(res.back().se, a[i].se);
    } else {
      res.push_back(a[i]);
    }
  }
  swap(a, res);
}
// firstの値が同じ場合secondが最大の物を残す
template <class T, class U> void unique_max(vector<pair<T, U>> &a) {
  vector<pair<T, U>> res;
  if (sz(a))
    res.push_back(a[0]);
  rep(i, 1, sz(a)) {
    if (res.back().fi == a[i].fi) {
      chma(res.back().se, a[i].se);
    } else {
      res.push_back(a[i]);
    }
  }
  swap(a, res);
}

template <class T> bool includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(c);
  sort(d);
  return includes(ALL(c), ALL(d));
}
template <class T> bool distinct(const vector<T> &A) {
  if ((int)(A).size() == 1)
    return true;
  if ((int)(A).size() == 2)
    return A[0] != A[1];
  if ((int)(A).size() == 3)
    return (A[0] != A[1] && A[1] != A[2] && A[0] != A[2]);
  auto B = A;
  sort(B);
  int N = (B.size());
  unique(B);
  return N == (int)(B.size());
}
template <class H, class... T> bool distinct(const H &a, const T &...b) {
  return distinct(vector<H>{a, b...});
}

/*@formatter:off*/
template <class T, class U> void inc(pair<T, U> &a, U v = 1) {
  a.first += v, a.second += v;
}
template <class T, class U> void inc(T &a, U v = 1) { a += v; }
template <class T, class U = int> void inc(vector<T> &a, U v = 1) {
  for (auto &u : a)
    inc(u, v);
}
template <class T, class U> void dec(T &a, U v = 1) { a -= v; }
template <class T, class U = int> void dec(vector<T> &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class U> void dec(string &a, U v = 1) {
  for (auto &u : a)
    dec(u, v);
}
template <class T, class U, class W>
void dec(vector<T> &a, vector<U> &b, W v = 1) {
  for (auto &u : a)
    dec(u, v);
  for (auto &u : b)
    dec(u, v);
}
template <class T, class U, class W>
void dec(vector<T> &a, vector<U> &b, vector<W> &c) {
  for (auto &u : a)
    dec(u, 1);
  for (auto &u : b)
    dec(u, 1);
  for (auto &u : c)
    dec(u, 1);
}
bool ins(ll h, ll w, ll H, ll W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool san(ll l, ll v, ll r) { return l <= v && v < r; }
template <class T> bool ins(vector<T> &a, ll i, ll j = 0) {
  return san(0, i, sz(a)) && san(0, j, sz(a));
}
#define inside ins
ll u0(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u0(vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) { v = u(v); }
  return ret;
}
// todo 名前
bool d_(int a, int b) {
  if (b == 0)
    return false;
  return (a % b) == 0;
}

// エラー
void ole() {
#ifdef _DEBUG
  cerr << "ole" << endl;
  exit(0);
#endif
  string a = "a";
  rep(i, 30) a += a;
  rep(i, 1 << 17) cout << a << endl;
  cout << "OLE 出力長制限超過" << endl;
  exit(0);
}
void re(string s = "") {
  cerr << s << endl;
  assert(0 == 1);
  exit(0);
}

void tle() {
  while (inf)
    cout << inf << endl;
}

//@汎用便利関数 入力
ll in() {
  ll ret;
  cin >> ret;
  return ret;
}
template <class T> T in() {
  T ret;
  cin >> ret;
  return ret;
}
string sin() {
  string ret;
  cin >> ret;
  return ret;
}
template <class T> void in(T &head) { cin >> head; }
template <class T, class... U> void in(T &head, U &...tail) {
  cin >> head;
  in(tail...);
}

// value_typeを持つ場合呼べる
// len回要素を追加する
template <class Iterable, class T = typename Iterable::value_type>
Iterable tin(int len) {
  Iterable ret;
  T tem;
  while (len--) {
    cin >> tem;
    ret += tem;
  }
  return ret;
}
template <class T> T tin() {
  T ret;
  cin >> ret;
  return ret;
}
template <class T> T tind(int len = 0) {
  auto ret = tin<T>(len);
  dec(ret, 1);
  return ret;
}
#define din_t2(type, a)                                                        \\
  type a;                                                                      \\
  cin >> a
#define din_t3(type, a, b)                                                     \\
  type a, b;                                                                   \\
  cin >> a >> b
#define din_t4(type, a, b, c)                                                  \\
  type a, b, c;                                                                \\
  cin >> a >> b >> c
#define din_t5(type, a, b, c, d)                                               \\
  type a, b, c, d;                                                             \\
  cin >> a >> b >> c >> d
#define din_t6(type, a, b, c, d, e)                                            \\
  type a, b, c, d, e;                                                          \\
  cin >> a >> b >> c >> d >> e
#define din_t7(type, a, b, c, d, e, f)                                         \\
  type a, b, c, d, e, f;                                                       \\
  cin >> a >> b >> c >> d >> e >> f
#define din_t(...)                                                             \\
  over7(__VA_ARGS__, din_t7, din_t6, din_t5, din_t4, din_t3,                   \\
        din_t2)(__VA_ARGS__)

#define din(...) din_t(int, __VA_ARGS__)
#define d_in
#define dsig(...) din_t(signed, __VA_ARGS__)

#define dst(...) din_t(string, __VA_ARGS__)
#define dstr dst
#define d_str dst

#define dcha(...) din_t(char, __VA_ARGS__)
#define dchar dcha

#define ddou(...) din_t(double, __VA_ARGS__)

#define din1d(a)                                                               \\
  din_t2(int, a);                                                              \\
  a--
#define din2d(a, b)                                                            \\
  din_t3(int, a, b);                                                           \\
  a--, b--
#define din3d(a, b, c)                                                         \\
  din_t4(int, a, b, c);                                                        \\
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \\
  din_t5(int, a, b, c, d);                                                     \\
  a--, b--, c--, d--
#define dind(...) over4(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

/*@formatter:off*/
#ifdef _DEBUG
template <class T> void err2(T &&head) { cerr << head; }
template <class T, class... U> void err2(T &&head, U &&...tail) {
  cerr << head << " ";
  err2(tail...);
}
template <class T, class... U> void err(T &&head, U &&...tail) {
  cerr << head << " ";
  err2(tail...);
  cerr << "" << endl;
}
template <class T> void err(T &&head) { cerr << head << endl; }
void err() { cerr << "" << endl; }
// debで出力する最大長
constexpr int DEB_LEN = 20;
constexpr int DEB_LEN_H = 12;
string deb_tos(const int &v) {
  if (abs(v) == inf || abs(v) == linf)
    return "e";
  else
    return to_string(v);
}
template <class T> string deb_tos(const T &a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
string deb_tos(const P &p) {
  stringstream ss;
  ss << "{";
  ss << p.fi;
  ss << ", ";
  ss << p.se;
  ss << "}";
  return ss.str();
}
#ifdef use_epsdou
string deb_tos(const epsdou &a) { return deb_tos(a.v); }
#endif
template <class T> string deb_tos(const optional<T> &a) {
  if (a.has_value()) {
    return deb_tos(a.value());
  } else
    return "e";
}
template <class T> string deb_tos(const vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), DEB_LEN);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) {
    ss << deb_tos(a[i]);
    if (typeid(a[i]) == typeid(P)) {
      ss << endl;
    } else {
      ss << " ";
    }
  }
  return ss.str();
}
template <class T>
string deb_tos(const vector<vector<T>> &a, vi H, vi W, int key = -1) {
  stringstream ss;
  ss << endl;
  vi lens(sz(W));
  fora(h, H) {
    rep(wi, sz(W)) {
      if (sz(a[h]) <= W[wi])
        break;
      lens[wi] = max(lens[wi], sz(deb_tos(a[h][W[wi]])) + 1);
      lens[wi] = max(lens[wi], sz(deb_tos(W[wi])) + 1);
    }
  }
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  int wi = 0;
  fora(w, W) {
    ss << std::right << std::setw(lens[wi]) << w;
    wi++;
  }
  ss << "" << endl;
  rep(i, sz(W)) rep(lens[i]) ss << "_";
  rep(i, 3) ss << "_";
  ss << "" << endl;
  fora(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    int wi = 0;
    fora(w, W) {
      if (sz(a[h]) <= w)
        break;
      ss << std::right << std::setw(lens[wi]) << deb_tos(a[h][w]);
      wi++;
    }
    ss << "" << endl;
  }
  return ss.str();
}
template <class T>
string deb_tos(const vector<vector<T>> &a, ll H = inf, ll W = inf,
               int key = -1) {
  H = (H != inf) ? H : min({H, sz(a), DEB_LEN_H});
  W = 0;
  rep(h, H) { chma(W, sz(a[h])); }
  chmi(W, DEB_LEN_H);
  vi hs, ws;
  rep(h, H) { hs.push_back(h); }
  rep(w, W) { ws.push_back(w); }
  return deb_tos(a, hs, ws, key);
}
/*@formatter:off*/
template <class T>
string deb_tos(const vector<vector<vector<T>>> &a, ll H = inf) {
  stringstream ss;
  if (H == inf)
    H = DEB_LEN_H;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << deb_tos(a[i], inf, inf, i);
  }
  return ss.str();
}
template <class T>
string deb_tos(vector<set<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  vector<vector<T>> b(sz(a));
  rep(i, sz(a)) {
    fora(v, a[i]) { b[i].push_back(v); }
  }
  return deb_tos(b, H, W, key);
}
template <class T>
string deb_tos(vector<multiset<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  vector<vector<T>> b(sz(a));
  rep(i, sz(a)) {
    fora(v, a[i]) { b[i].push_back(v); }
  }
  return deb_tos(b, H, W, key);
}
template <class T> string deb_tos(const set<T> &S) {
  vector<T> b;
  fora(s, S) b += s;
  return deb_tos(b);
}
template <class T, size_t A> string deb_tos(T (&a)[A]) {
  return deb_tos(vector<T>(begin(a), end(a)));
}
template <class T, size_t A, size_t B> string deb_tos(T (&a)[A][B]) {
  return deb_tos(vector<vector<T>>(begin(a), end(a)));
}
template <class T, size_t A, size_t B, size_t C>
string deb_tos(T (&a)[A][B][C]) {
  return deb_tos(vector<vector<vector<T>>>(begin(a), end(a)));
}
/*@formatter:off*/
template <class T> void out2(T head) {
  cout << head;
  res_mes += deb_tos(head);
}
template <class T, class... U> void out2(T head, U... tail) {
  cout << head << " ";
  res_mes += deb_tos(head) + " ";
  out2(tail...);
}
template <class T, class... U> void out(T head, U... tail) {
  cout << head << " ";
  res_mes += deb_tos(head) + " ";
  out2(tail...);
  cout << "" << endl;
  res_mes += "\\n";
}
template <class T> void out(T head) {
  cout << head << endl;
  res_mes += deb_tos(head) + "\\n";
}
void out() { cout << "" << endl; }

#else
#define err(...) ;
template <class T> void out2(T &&head) { cout << head; }
template <class T, class... U> void out2(T &&head, U &&...tail) {
  cout << head << " ";
  out2(tail...);
}
template <class T, class... U> void out(T &&head, U &&...tail) {
  cout << head << " ";
  out2(tail...);
  cout << "" << endl;
}
template <class T> void out(T &&head) { cout << head << endl; }
void out() { cout << "" << endl; }

#endif

template <class H, class... T>
void outl(const vector<H> &h, const vector<T> &...t) {
  rep(i, sz(h)) out(h[i], t[i]...);
}

// テーブルをスペースなしで出力
template <class T> void outt(vector<vector<T>> &a) {
  rep(i, sz(a)) {
    rep(j, sz(a[i])) { cout << a[i][j]; }
    cout << endl;
  }
}
// int型をbit表記で出力
void outb(int a) { cout << bitset<20>(a) << endl; }

/*@formatter:off*/
template <class T> void na(vector<T> &a, ll n) {
  a.resize(n);
  rep(i, n) cin >> a[i];
}
template <class T> void na(set<T> &a, ll n) { rep(i, n) a.insert(in()); }
#define dna(a, n)                                                              \\
  vi a;                                                                        \\
  na(a, n); /*nを複数使うと n==in()の時バグる事に注意*/
#define dnad(a, n)                                                             \\
  vi a;                                                                        \\
  nad(a, n);
template <class T> void nao(vector<T> &a, ll n) {
  a.resize(n + 1);
  a[0] = 0;
  rep(i, n) cin >> a[i + 1];
}
template <class T> void naod(vector<T> &a, ll n) {
  a.resize(n + 1);
  a[0] = 0;
  rep(i, n) cin >> a[i + 1], a[i + 1]--;
}
template <class T> void nad(vector<T> &a, ll n) {
  a.resize(n);
  rep(i, n) cin >> a[i], a[i]--;
}
template <class T> void nad(set<T> &a, ll n) { rep(i, n) a.insert(in() - 1); }
template <class T, class U> void na2(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n);
  b.resize(n);
  rep(i, n) cin >> a[i] >> b[i];
}
template <class T, class U> void na2(set<T> &a, set<U> &b, ll n) {
  rep(i, n) {
    a.insert(in());
    b.insert(in());
  }
}
#define dna2(a, b, n)                                                          \\
  vi a, b;                                                                     \\
  na2(a, b, n);
template <class T, class U> void nao2(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n + 1);
  b.resize(n + 1);
  a[0] = b[0] = 0;
  rep(i, n) cin >> a[i + 1] >> b[i + 1];
}
#define dnao2(a, b, n)                                                         \\
  vi a, b;                                                                     \\
  nao2(a, b, n);
template <class T, class U> void na2d(vector<T> &a, vector<U> &b, ll n) {
  a.resize(n);
  b.resize(n);
  rep(i, n) cin >> a[i] >> b[i], a[i]--, b[i]--;
}
#define dna2d(a, b, n)                                                         \\
  vi a, b;                                                                     \\
  na2d(a, b, n);
template <class T, class U, class W>
void na3(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i];
}
#define dna3(a, b, c, n)                                                       \\
  vi a, b, c;                                                                  \\
  na3(a, b, c, n);
template <class T, class U, class W>
void na3d(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;
}
#define dna3d(a, b, c, n)                                                      \\
  vi a, b, c;                                                                  \\
  na3d(a, b, c, n);
template <class T, class U, class W, class X>
void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, ll n) {
  a.resize(n);
  b.resize(n);
  c.resize(n);
  d.resize(n);
  rep(i, n) cin >> a[i] >> b[i] >> c[i] >> d[i];
}
#define dna4(a, b, c, d, n)                                                    \\
  vi a, b, c, d;                                                               \\
  na4(a, b, c, d, n);
#define dna4d(a, b, c, d, n)                                                   \\
  vi a, b, c, d;                                                               \\
  na4d(a, b, c, d, n);
#define nt(a, h, w)                                                            \\
  resize(a, h, w);                                                             \\
  rep(nthi, h) rep(ntwi, w) cin >> a[nthi][ntwi];
#define ntd(a, h, w)                                                           \\
  resize(a, h, w);                                                             \\
  rep(ntdhi, h) rep(ntdwi, w) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w)                                                           \\
  resize(a, h + 2, w + 2);                                                     \\
  fill(a, '#');                                                                \\
  rep(ntphi, 1, h + 1) rep(ntpwi, 1, w + 1) cin >> a[ntphi][ntpwi];
#define dnt(S, h, w)                                                           \\
  vvi(S, h, w);                                                                \\
  nt(S, h, w);
#define dntc(S, h, w)                                                          \\
  vvc(S, h, w);                                                                \\
  nt(S, h, w);
#define dnts(S, h, w)                                                          \\
  vvs(S, h, w);                                                                \\
  nt(S, h, w);

// デバッグ
#define sp << " " <<
/*@formatter:off*/
#define deb1(x) debugName(x) << " = " << deb_tos(x)
#define deb_2(x, ...) deb1(x) << ", " << deb1(__VA_ARGS__)
#define deb_3(x, ...) deb1(x) << ", " << deb_2(__VA_ARGS__)
#define deb_4(x, ...) deb1(x) << ", " << deb_3(__VA_ARGS__)
#define deb5(x, ...) deb1(x) << ", " << deb_4(__VA_ARGS__)
#define deb6(x, ...) deb1(x) << ", " << deb5(__VA_ARGS__)
// #define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)
// #define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)
// #define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)
// #define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)

/*@formatter:off*/
#ifdef _DEBUG
#define deb(...)                                                               \\
  do {                                                                         \\
    was_deb = true;                                                            \\
    cerr << over10(__VA_ARGS__, deb10, deb9, deb8, deb7, deb6, deb5, deb_4,    \\
                   deb_3, deb_2, deb1)(__VA_ARGS__)                            \\
         << endl;                                                              \\
  } while (0)

void print_n_base(int x, int base) {
  string S;
  while (x) {
    S += (char)('0' + x % base);
    x /= base;
  }
  cerr << S << endl;
}
template <class T> void print_n_base(vector<T> X, int base) {
  cerr << endl;
  for (auto &&x : X) {
    print_n_base(x, base);
  }
  cerr << endl;
}

// n進数
#define deb2(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 2);
#define deb3(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 3);
#define deb4(x)                                                                \\
  was_deb = true;                                                              \\
  cerr << debugName(x) << " = ";                                               \\
  print_n_base(x, 4);

#define deb_ex_deb(x, len) debugName(x) << " = " << deb_tos(x, len)
#define call_deb_ex_deb(x, len) deb_ex_deb(x, len)
// 要素が存在する行だけ出力(vvt)
#define deb_ex(v)                                                              \\
  do {                                                                         \\
    int N = sz(v);                                                             \\
    int s = N;                                                                 \\
    int t = 0;                                                                 \\
    rep(i, N) {                                                                \\
      if (sz(v[i])) {                                                          \\
        chmi(s, i);                                                            \\
        chma(t, i);                                                            \\
      }                                                                        \\
    }                                                                          \\
    auto ex_v = sub(v, s, N);                                                  \\
    str S = deb_tos(ex_v, sz(ex_v));                                           \\
    debugName(v);                                                              \\
    cerr << " = " << endl;                                                     \\
    cerr << S << endl;                                                         \\
  } while (0);
#define debi(A)                                                                \\
  {                                                                            \\
    int len = min(sz(A), 20);                                                  \\
    was_deb = true;                                                            \\
    cerr << debugName(A) << " = " << endl;                                     \\
    rep(i, len) cerr << std::right                                             \\
                     << std::setw((int)(sz(tos(A[i])) + (i ? 1 : 0)))          \\
                     << (i % 10);                                              \\
    cerr << endl;                                                              \\
    rep(i, len) cerr << std::right                                             \\
                     << std::setw((int)(sz(tos(A[i])) + (i ? 1 : 0))) << A[i]; \\
    cerr << endl;                                                              \\
  }
template <class T, class F> string deb_tos_f(vector<T> &A, F f) {
  vi I;
  rep(i, sz(A)) {
    if (f(A[i]))
      I.push_back(i);
  }
  was_deb = true;
  stringstream ss;
  fora(ii, i, I) ss << std::right
                    << std::setw(
                           (int)(max(sz(tos(A[i])), sz(tos(i))) + (ii ? 1 : 0)))
                    << i;
  ss << endl;
  fora(ii, i, I) ss << std::right
                    << std::setw(
                           (int)(max(sz(tos(A[i])), sz(tos(i))) + (ii ? 1 : 0)))
                    << A[i];
  return ss.str();
}
template <class T, class F>
string deb_tos_f(vector<vector<T>> &a, F f, int key = -1) {
  vi hs, ws_;
  int H = sz(a), W = sz(a[0]);
  vi exh(H), exw(W);
  rep(h, H) {
    rep(w, W) {
      if (f(a[h][w])) {
        exh[h] = true;
        exw[w] = true;
      }
    }
  }
  rep(h, H) if (exh[h]) hs.push_back(h);
  rep(w, W) if (exw[w]) ws_.push_back(w);
  return deb_tos(a, hs, ws_, key);
}
template <class T, class F>
string deb_tos_f(vector<vector<vector<T>>> &a, F f) {
  stringstream ss;
  int H = sz(a);
  if (sz(a) == 0)
    return ss.str();
  int cou = 0;
  rep(i, H) {
    if (any_of(a[i], v, f(v))) {
      ss << deb_tos_f(a[i], f, i);
      cou++;
    }
    if (cou > 50)
      break;
  }
  ss << "" << endl;
  return ss.str();
}
#define debf_normal(tab, f)                                                    \\
  do {                                                                         \\
    cerr << debugName(tab) << " = " << endl;                                   \\
    cerr << deb_tos_f(tab, f) << endl;                                         \\
  } while (0);
#define debf2(tab, siki_r) debf_normal(tab, lamr(siki_r))
#define debf3(tab, v, siki) debf_normal(tab, lam(siki))
// S, sikir
// S, v, siki
#define debf(...) over3(__VA_ARGS__, debf3, debf2, debf1)(__VA_ARGS__)

#else
#define deb(...) ;
#define deb2(...) ;
#define deb3(...) ;
#define deb4(...) ;
#define deb_ex(...) ;
#define debf(...) ;
#define debi(...) ;
#endif

#define debugline(x)                                                           \\
  cerr << x << " "                                                             \\
       << "(L:" << __LINE__ << ")" << '\\n'

/*@formatter:off*/
using u32 = unsigned;
using u64 = unsigned long long;
using u128 = __uint128_t;

using bint = __int128;
std::ostream &operator<<(std::ostream &dest, __int128_t value) {
  std::ostream::sentry s(dest);
  if (s) {
    __uint128_t tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = std::end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    ll len = std::end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(std::ios_base::badbit);
    }
  }
  return dest;
}
__int128 to_bint(string &s) {
  __int128 ret = 0;
  for (ll i = 0; i < (ll)s.length(); ++i)
    if ('0' <= s[i] && s[i] <= '9')
      ret = 10 * ret + s[i] - '0';
  return ret;
}
void operator>>(istream &iss, bint &v) {
  string S;
  iss >> S;
  v = 0;
  rep(i, sz(S)) {
    v *= 10;
    v += S[i] - '0';
  }
}

// 便利関数
/*@formatter:off*/
// テスト用
#define rand xor128_
unsigned long xor128_(void) {
  static unsigned long x = 123456789, y = 362436069, z = 521288629,
                       w = 88675123;
  unsigned long t;
  t = (x ^ (x << 11));
  x = y;
  y = z;
  z = w;
  return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
}
char ranc() { return (char)('a' + rand() % 26); }
ll rand(ll min, ll max) {
  assert(min <= max);
  if (min >= 0 && max >= 0) {
    return rand() % (max + 1 - min) + min;
  } else if (max < 0) {
    return -rand(-max, -min);
  } else {
    if (rand() % 2) {
      return rand(0, max);
    } else {
      return -rand(0, -min);
    }
  }
}
ll rand(ll max) { return rand(0, max); }
template <class T> T rand(vector<T> &A) { return A[rand(sz(A) - 1)]; }
// 重複することがある
template <class T> vector<T> ranv(vector<T> &A, int N) {
  vector<T> ret(N);
  rep(i, N) { ret[i] = rand(A); }
  return ret;
}
template <class T> vector<T> ranv_unique(vector<T> &A, int N) {
  vector<T> ret(N);
  umapi was;
  rep(j, N) {
    int i;
    while (1) {
      i = rand(sz(A) - 1);
      if (was.find(i) == was.end())
        break;
    }
    ret[j] = A[i];
    was[i] = 1;
  }
  return ret;
}
vi ranv(ll n, ll min, ll max) {
  vi v(n);
  rep(i, n) v[i] = rand(min, max);
  return v;
}

/*@formatter:off*/
#ifdef _DEBUG
bool timeup(int time) {
  static bool never = true;
  if (never)
    message += "may timeup, because slow";
  never = false;
  auto end_time = system_clock::now();
  auto part = duration_cast<milliseconds>(end_time - start_time);
  auto lim = milliseconds(time);
  return part >= lim;
}
#else
bool timeup(int time) {
  auto end_time = system_clock::now();
  auto part = duration_cast<milliseconds>(end_time - start_time);
  auto lim = milliseconds(time);
  return part >= lim;
}

#endif
void set_time() { past_time = system_clock::now(); }
// MS型(millisecqnds)で返る
// set_timeをしてからの時間
auto calc_time_milli() {
  auto now = system_clock::now();
  auto part = duration_cast<milliseconds>(now - past_time);
  return part;
}
auto calc_time_micro() {
  auto now = system_clock::now();
  auto part = duration_cast<microseconds>(now - past_time);
  return part;
}
auto calc_time_nano() {
  auto now = system_clock::now();
  auto part = duration_cast<nanoseconds>(now - past_time);
  return part;
}
bool calc_time(int zikan) { return calc_time_micro() >= microseconds(zikan); }
using MS = std::chrono::microseconds;
int div(microseconds a, microseconds b) { return a / b; }
int div(nanoseconds a, nanoseconds b) {
  if (b < nanoseconds(1)) {
    return a / nanoseconds(1);
  }
  int v = a / b;
  return v;
}
// set_time();
// rep(i,lim)shori

// lim*=time_nanbai();
// rep(i,lim)shoriと使う
// 全体でmilliかかっていいときにlimを何倍してもう一回できるかを返す
int time_nanbai(int milli) {
  auto dec = duration_cast<nanoseconds>(past_time - start_time);
  auto part = calc_time_nano();
  auto can_time = nanoseconds(milli * 1000 * 1000);
  can_time -= part;
  can_time -= dec;
  return div(can_time, part);
}

/*@formatter:off*/
// #define use_rand
#ifdef use_rand
str ransu(ll n) {
  str s;
  rep(i, n) s += (char)rand('A', 'Z');
  return s;
}
str ransl(ll n) {
  str s;
  rep(i, n) s += (char)rand('a', 'z');
  return s;
}
// 単調増加
vi ranvinc(ll n, ll min, ll max) {
  vi v(n);
  bool bad = 1;
  while (bad) {
    bad = 0;
    v.resize(n);
    rep(i, n) {
      if (i && min > max - v[i - 1]) {
        bad = 1;
        break;
      }
      if (i)
        v[i] = v[i - 1] + rand(min, max - v[i - 1]);
      else
        v[i] = rand(min, max);
    }
  }
  return v;
}
// 便利 汎用
#endif

void ranvlr(ll n, ll min, ll max, vi &l, vi &r) {
  l.resize(n);
  r.resize(n);
  rep(i, n) {
    l[i] = rand(min, max);
    r[i] = l[i] + rand(0, max - l[i]);
  }
}
template <class Iterable, class T = typename Iterable::value_type>
vector<pair<T, int>> run_length(const Iterable &a) {
  vector<pair<T, int>> ret;
  ret.eb(a[0], 1);
  rep(i, 1, sz(a)) {
    if (ret.back().fi == a[i]) {
      ret.back().se++;
    } else {
      ret.eb(a[i], 1);
    }
  }
  return ret;
}

/*@formatter:off*/
// #define use_mgr //_goldd _goldt
#ifdef use_mgr
//->[i, f(i)]
template <class T, class U, class F>
auto mgr(T ok, U ng, const F &f,
         require_arg(is_integral<T>::value &&is_integral<U>::value)) {
  auto mid = (ok + ng);
  if (ok < ng)
    while (ng - ok > 1) {
      mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  else
    while (ok - ng > 1) {
      mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  return ok;
}
//[l, r)の中で,f(i)がtrueとなる範囲を返す okはそこに含まれる
template <class F> P mgr_range(int l, int r, F f, int ok) {
  if (f(ok) == 0) {
    out("f(ok) must true");
    re();
  }
  return mp(mgr(ok, l - 1, f), mgr(ok, r, f) + 1);
}
template <class F> auto mgrd(dou ok, dou ng, F f, int kai = 100) {
  if (ok < ng)
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  else
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid;
      else
        ng = mid;
    }
  return ok;
}
template <class F> dou mgrd_time(dou ok, dou ng, F f, int time = 1980) {
  bool han = true;
  if (ok < ng)
    while (1) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
      if (timeup(time)) {
        break;
      }
    }
  else
    while (1) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
      if (timeup(time)) {
        break;
      }
    }
  return ok;
}
// todo 減らす
template <class F> auto goldd_l(ll left, ll right, F calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  ll minScore = MAX<ll>();
  ll resIndex = left;
  for (ll i = left; i < right + 1; ++i) {
    ll score = calc(i);
    if (minScore > score) {
      minScore = score;
      resIndex = i;
    }
  }
  return make_tuple(resIndex, calc(resIndex));
}
template <class F> auto goldt_l(ll left, ll right, F calc) {
  double GRATIO = 1.6180339887498948482045868343656;
  ll lm = left + (ll)((right - left) / (GRATIO + 1.0));
  ll rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
  ll fl = calc(lm);
  ll fr = calc(rm);
  while (right - left > 10) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + (ll)((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + (ll)((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  if (left > right) {
    ll l = left;
    left = right;
    right = l;
  }
  ll maxScore = MIN<ll>();
  ll resIndex = left;
  for (ll i = left; i < right + 1; ++i) {
    ll score = calc(i);
    if (maxScore < score) {
      maxScore = score;
      resIndex = i;
    }
  }
  return make_tuple(resIndex, calc(resIndex));
}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template <class F> auto goldd_d(dou left, dou right, F calc, ll loop = 200) {
  dou GRATIO = 1.6180339887498948482045868343656;
  dou lm = left + ((right - left) / (GRATIO + 1.0));
  dou rm = lm + ((right - lm) / (GRATIO + 1.0));
  dou fl = calc(lm);
  dou fr = calc(rm); /*200にすればおそらく大丈夫*/ /*余裕なら300に*/
  ll k = 141;
  loop++;
  while (--loop) {
    if (fl < fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + ((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + ((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  return make_tuple(left, calc(left));
}
template <class F> auto goldt_d(dou left, dou right, F calc, ll loop = 200) {
  double GRATIO = 1.6180339887498948482045868343656;
  dou lm = left + ((right - left) / (GRATIO + 1.0));
  dou rm = lm + ((right - lm) / (GRATIO + 1.0));
  dou fl = calc(lm);
  dou fr = calc(rm);
  loop++;
  while (--loop) {
    if (fl > fr) {
      right = rm;
      rm = lm;
      fr = fl;
      lm = left + ((right - left) / (GRATIO + 1.0));
      fl = calc(lm);
    } else {
      left = lm;
      lm = rm;
      fl = fr;
      rm = lm + ((right - lm) / (GRATIO + 1.0));
      fr = calc(rm);
    }
  }
  return make_tuple(left, calc(left));
}
// l ~ rを複数の区間に分割し、極致を与えるiを返す time-20 msまで探索
template <class F> auto goldd_ls(ll l, ll r, F calc, ll time = 2000) {
  auto lim = milliseconds(time - 20);
  ll mini = 0, minv = MAX<ll>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    ll haba = (r - l + k) / k; /*((r-l+1) + k-1) /k*/
    ll nl = l;
    ll nr = l + haba;
    rep(i, k) {
      ll ni = goldd_l(nl, nr, calc);
      if (chmi(minv, calc(ni)))
        mini = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(mini, calc(mini));
}
template <class F> auto goldt_ls(ll l, ll r, F calc, ll time = 2000) {
  auto lim = milliseconds(time - 20);
  ll maxi = 0, maxv = MIN<ll>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    ll haba = (r - l + k) / k; /*((r-l+1) + k-1) /k*/
    ll nl = l;
    ll nr = l + haba;
    rep(i, k) {
      ll ni = goldt_l(nl, nr, calc);
      if (chma(maxv, calc(ni)))
        maxi = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(maxi, calc(maxi));
}
template <class F>
auto goldd_d_s(dou l, dou r, F calc, ll time = 2000) { /*20ms余裕を持つ*/
  auto lim = milliseconds(time - 20);
  dou mini = 0, minv = MAX<dou>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    dou haba = (r - l) / k;
    dou nl = l;
    dou nr = l + haba;
    rep(i, k) {
      dou ni = goldd_d(nl, nr, calc);
      if (chmi(minv, calc(ni)))
        mini = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(mini, calc(mini));
}
template <class F>
auto goldt_d_s(dou l, dou r, F calc, ll time = 2000) { /*20ms余裕を残している*/
  auto lim = milliseconds(time - 20);
  dou maxi = 0, maxv = MIN<dou>(); /*区間をk分割する*/
  rep(k, 1, inf) {
    auto s = system_clock::now();
    dou haba = (r - l) / k;
    dou nl = l;
    dou nr = l + haba;
    rep(i, k) {
      dou ni = goldt_d(nl, nr, calc);
      if (chma(maxv, calc(ni)))
        maxi = ni;
      nl = nr;
      nr = nl + haba;
    }
    auto end = system_clock::now();
    auto part = duration_cast<milliseconds>(end - s);
    auto elapsed = duration_cast<milliseconds>(end - start_time);
    if (elapsed + part * 2 >= lim) {
      break;
    }
  }
  return make_tuple(maxi, calc(maxi));
}
#endif

// strを整数として比較
string smax(str &a, str b) {
  if (sz(a) < sz(b)) {
    return b;
  } else if (sz(a) > sz(b)) {
    return a;
  } else if (a < b)
    return b;
  else
    return a;
}
// strを整数として比較
string smin(str &a, str b) {
  if (sz(a) > sz(b)) {
    return b;
  } else if (sz(a) < sz(b)) {
    return a;
  } else if (a > b)
    return b;
  else
    return a;
}
// エラーN
// 多次元なら(-1,-1,-1)
template <typename W, typename T> ll find(vector<W> &a, int l, const T key) {
  rep(i, l, sz(a)) if (a[i] == key) return i;
  return sz(a);
}
template <typename W, typename T> ll find(vector<W> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return sz(a);
}
template <typename W, typename T> P find(vector<vector<W>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return mp(i, j);
  return mp(-1, -1);
}
// getid(find())を返す 1次元にする
template <typename W, typename T> int findi(vector<vector<W>> &a, const T key) {
  rep(i, sz(a)) rep(j, sz(a[0])) if (a[i][j] == key) return i * sz(a[0]) + j;
  return -1;
}
template <typename W, typename U>
tuple<int, int, int> find(vector<vector<vector<W>>> &a, const U key) {
  rep(i, sz(a)) rep(j, sz(a[0]))
      rep(k, sz(a[0][0])) if (a[i][j][k] ==
                              key) return tuple<int, int, int>(i, j, k);
  return tuple<int, int, int>(-1, -1, -1);
}

// 無ければN
int find(string &s, const string key) {
  int klen = sz(key);
  rep(i, sz(s) - klen + 1) {
    if (s[i] != key[0])
      continue;
    if (s.substr(i, klen) == key) {
      return i;
    }
  }
  return sz(s);
}
int find(string &s, int l, const string key) {
  int klen = sz(key);
  rep(i, l, sz(s) - klen + 1) {
    if (s[i] != key[0])
      continue;
    if (s.substr(i, klen) == key) {
      return i;
    }
  }
  return sz(s);
}
int find(string &s, const char key) {
  rep(i, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return sz(s);
}
int find(string &s, int l, const char key) {
  rep(i, l, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return sz(s);
}

// N箇所について(自分含む)右のkeyの場所を返す
template <typename W, typename T> vi finds(const W &a, const T &key) {
  int n = sz(a);
  vi rpos(n, n);
  rer(i, n - 1) {
    if (i < n - 1) {
      rpos[i] = rpos[i + 1];
    }
    if (a[i] == key)
      rpos[i] = i;
  }
  return rpos;
}
template <typename W, typename T> vi rfinds(const W &a, const T &key) {
  int n = sz(a);
  vi lpos(n, -1);
  rep(i, n) {
    if (i > 0) {
      lpos[i] = lpos[i - 1];
    }
    if (a[i] == key)
      lpos[i] = i;
  }
  return lpos;
}
// todoz
#if __cplusplus >= 201703L
template <typename W, typename T, class Iterable = typename W::value_type>
ll count(const W &a, const T &k) {
  return count_if(a, == k);
}
template <typename W, class Iterable = typename W::value_type>
vi count(const W &a) {
  vi res;
  for_each(a, v, if (sz(res) <= (int)v) res.resize((int)v + 1); res[v]++;);
  return res;
}
#else
ll count(const string &S, const char &c) {
  int res = 0;
  fora(s, S) { res += s == c; }
  return res;
}
#endif
ll count(const str &a, const str &k) {
  ll ret = 0, len = k.length();
  auto pos = a.find(k);
  while (pos != string::npos)
    pos = a.find(k, pos + len), ++ret;
  return ret;
}
/*@formatter:off*/
//'a' = 'A' = 0 として集計 既に-'a'されていても動く
vi count(str &a, int l, int r) {
  vi cou(26);
  char c = 'a';
  if ('A' <= a[l] && a[l] <= 'Z')
    c = 'A';
  if ('a' <= a[l] && a[l] <= 'z')
    c = 'a';
  else
    c = 0;
  rep(i, l, r)++ cou[a[i] - c];
  return cou;
}
#define couif count_if
// algorythm
ll rev(ll a) {
  ll res = 0;
  while (a) {
    res *= 10;
    res += a % 10;
    a /= 10;
  }
  return res;
}
template <class T> auto rev(const vector<T> &a) {
  auto b = a;
  reverse(ALL(b));
  return b;
}
/* ＼反転  */ template <class U> auto rev(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a[0]), vector<U>(sz(a)));
  rep(h, sz(a)) rep(w, sz(a[0])) b[w][h] = a[h][w];
  return b;
}
/* ｜反転  */ template <class U> auto revw(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a), vector<U>(sz(a[0])));
  int W = sz(a[0]);
  rep(h, sz(a)) rep(w, sz(a[0])) { b[h][W - 1 - w] = a[h][w]; }
  return b;
}

/* ー反転  */ template <class U> auto revh(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a), vector<U>(sz(a[0])));
  int H = sz(a);
  rep(h, sz(a)) rep(w, sz(a[0])) { b[H - 1 - h][w] = a[h][w]; }
  return b;
}

/* ／反転  */ template <class U> auto revr(vector<vector<U>> &a) {
  vector<vector<U>> b(sz(a[0]), vector<U>(sz(a)));
  int H = sz(a);
  int W = sz(a[0]);
  rep(h, sz(a)) rep(w, sz(a[0])) b[w][h] = a[H - 1 - h][W - 1 - w];
  return b;
}

auto rev(const string &a) {
  string b = a;
  reverse(ALL(b));
  return b;
}
template <class T> auto rev(const T &v, int i) { return v[sz(v) - 1 - i]; }
int rev(int N, int i) { return N - 1 - i; }
constexpr ll p10[] = {1,
                      10,
                      100,
                      1000,
                      10000,
                      100000,
                      1000000,
                      10000000,
                      100000000,
                      1000000000,
                      10000000000ll,
                      100000000000ll,
                      1000000000000ll,
                      10000000000000ll,
                      100000000000000ll,
                      1000000000000000ll,
                      10000000000000000ll,
                      100000000000000000ll,
                      1000000000000000000ll};

// 0は0桁
ll keta(ll v, int if_zero_res) {
  if (!v)
    return if_zero_res;
  if (v < p10[9]) {
    if (v < p10[4]) {
      if (v < p10[2]) {
        if (v < p10[1]) {
          if (v < p10[0])
            return 0;
          else
            return 1;
        } else
          return 2;
      } else {
        if (v < p10[3])
          return 3;
        else
          return 4;
      }
    } else {
      if (v < p10[7]) {
        if (v < p10[5])
          return 5;
        else if (v < p10[6])
          return 6;
        else
          return 7;
      } else {
        if (v < p10[8])
          return 8;
        else
          return 9;
      }
    }
  } else {
    if (v < p10[13]) {
      if (v < p10[11]) {
        if (v < p10[10])
          return 10;
        else
          return 11;
      } else {
        if (v < p10[12])
          return 12;
        else
          return 13;
      }
    } else {
      if (v < p10[15]) {
        if (v < p10[14])
          return 14;
        else
          return 15;
      } else {
        if (v < p10[17]) {
          if (v < p10[16])
            return 16;
          else
            return 17;
        } else {
          if (v < p10[18])
            return 18;
          else
            return 19;
        }
      }
    }
  }
}
#if __cplusplus >= 201703L
ll getr(ll a, ll keta) { return (a / pow<ll>(10, keta)) % 10; }
#else
ll getr(ll a, ll keta) { return (a / (int)pow(10, keta)) % 10; }
#endif
// 上から何桁目か
ll getl(ll a, ll ket) {
  int sketa = keta(a, 1);
  return getr(a, sketa - 1 - ket);
}
ll dsum(ll v, ll sin = 10) {
  ll ret = 0;
  for (; v; v /= sin)
    ret += v % sin;
  return ret;
}

ll mask10(ll v) { return p10[v] - 1; }
// 変換系

template <class T, class U>
auto to_v1(vector<reference_wrapper<U>> &ret, vector<T> &A) {
  rep(i, sz(A)) ret.push_back(A[i]);
  return ret;
}
template <class T, class U>
auto to_v1(vector<reference_wrapper<U>> &ret, vector<vector<T>> &A) {
  rep(i, sz(A)) to_v1(ret, A[i]);
  return ret;
}
// 参照付きで1次元に起こす
template <class T> auto to_v1(vector<vector<T>> &A) {
  vector<reference_wrapper<decl2_t<decltype(A)>>> ret;
  rep(i, sz(A)) to_v1(ret, A[i]);
  return ret;
}

//[v] := iとなるようなvectorを返す
// 存在しない物は-1
// 空でも動く(なぜか)
template <class T> auto keys(const T &a) {
  vector<decltype((a.begin())->fi)> res;
  for (auto &&k : a)
    res.push_back(k.fi);
  return res;
}
template <class T> auto values(const T &a) {
  vector<decltype((a.begin())->se)> res;
  for (auto &&k : a)
    res.push_back(k.se);
  return res;
}
// todo 可変長で
template <class T> constexpr T min(T a, T b, T c) {
  return a >= b ? b >= c ? c : b : a >= c ? c : a;
}
template <class T> constexpr T max(T a, T b, T c) {
  return a <= b ? b <= c ? c : b : a <= c ? c : a;
}
// 1次元のコンテナ
template <class T> int mini(const vector<T> &a) {
  return min_element(ALL(a)) - a.begin();
}
template <class T> int maxi(const vector<T> &a) {
  return max_element(ALL(a)) - a.begin();
}
template <class T> T sum(const T &A) { return A; }
template <class T> T sum(const vector<T> &A, int l = -1, int r = -1) {
  T s = 0;
  set_lr12(l, r, sz(A));
  rep(i, l, r) s += sum(A[i]);
  return s;
}
// 1次元のIterableに使える
template <class T, require_t(!has_value_type<T>::value)> T min(T &a) {
  return a;
}
template <class V, require_t(has_value_type<V>::value)>
auto min(V &a, ll s = -1, ll n = -1) {
  using Ret_t = decl2_t<V>;
  auto ret = MAX<Ret_t>();
  set_lr12(s, n, sz(a));
  if (s == n) {
    return INF<Ret_t>();
  } else {
    rep(i, s, n) { ret = min(ret, min(a[i])); }
    return ret;
  }
}
template <class T, require_t(!has_value_type<T>::value)> T max(T &a) {
  return a;
}
template <class V, require_t(has_value_type<V>::value)>
auto max(V &a, ll s = -1, ll n = -1) {
  using Ret_t = decl2_t<V>;
  auto ret = MIN<Ret_t>();
  set_lr12(s, n, sz(a));
  if (s == n) {
    return -INF<Ret_t>();
  } else {
    rep(i, s, n) { ret = max(ret, max(a[i])); }
    return ret;
  }
}
template <class T> T mul(vector<T> &v, ll t = inf) {
  T ret = v[0];
  rep(i, 1, min(t, sz(v))) ret *= v[i];
  return ret;
}
// template<class T, class U, class... W>  auto sumn(vector<T> &v, U head, W...
// tail) {    auto ret = sum(v[0], tail...);    rep(i, 1, min(sz(v), head))ret
// += sum(v[i], tail...);    return ret;} indexを持つvectorを返す

/*@formatter:off*/
template <typename W, typename T> void fill(W &xx, const T vall) { xx = vall; }
template <typename W, typename T> void fill(vector<W> &vecc, const T vall) {
  for (auto &&vx : vecc)
    fill(vx, vall);
}
template <typename W, typename T> void fill(vector<W> &xx, const T v, ll len) {
  rep(i, len) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<W> &xx, const T v, int s, ll t) {
  rep(i, s, t) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<vector<W>> &xx, T v, int sh, int th, int sw, int tw) {
  rep(h, sh, th) rep(w, sw, tw) xx[h][w] = v;
}
// #define use_fill //_sum _array _max _min
#ifdef use_fill
template <typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) {
  rep(i, N) { fill(a[i], v); }
}
template <class T, class U> void fill(vector<T> &a, const vi &ind, U val) {
  fora(v, ind) { a[v] = val; }
}
template <typename A, size_t N> auto sum(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = sum(a[l]);
  if (l == r) {
    return res - res;
  }
  rep(i, l + 1, r) res += sum(a[i]);
  return res;
}
template <typename A, size_t N> auto max(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = max(a[l]);
  if (l == r) {
    return -INF<decltype(res)>();
  }
  rep(i, l + 1, r) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> auto min(A (&a)[N], int l = -1, int r = -1) {
  set_lr12(l, r, N);
  auto res = min(a[l]);
  if (l == r) {
    return INF<decltype(res)>();
  }
  rep(i, l + 1, r) res = min(res, min(a[i]));
  return res;
}
#endif

vi inds_(vi &a) {
  int n = max(a) + 1;
  vi ret(n, -1);
  rep(i, sz(a)) {
    assert(ret[a[i]] == -1);
    ret[a[i]] = i;
  }
  return ret;
}
void clear(PQ &q) { q = PQ(); }
void clear(priority_queue<int> &q) { q = priority_queue<int>(); }
template <class T> void clear(queue<T> &q) {
  while (q.size())
    q.pop();
}
// template<class T> T *negarr(ll size) {    T *body = (T *) malloc((size * 2 +
// 1) * sizeof(T));    return body + size;} template<class T> T *negarr2(ll h,
// ll w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new
// double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (ll i =
// 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }    double
// **a = dummy1 + h;    return a;}

template <class T> struct ruiC {
  vector<T> rui;
  ruiC(vector<T> &ru) : rui(ru) {}
  /*先頭0*/
  ruiC() : rui(1, 0) {}
  T operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruic ";
      deb(l, r);
      assert(0);
    }
    return rui[r] - rui[l];
  }
  T operator()(int r = inf) { return operator()(0, min(r, sz(rui) - 1)); }
  /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
  /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
  //    T operator[](ll i) { return rui[i + 1] - rui[i]; }
  T operator[](ll i) { return rui[i]; }
  /*0から順に追加される必要がある*/
  void operator+=(T v) { rui.push_back(rui.back() + v); }
  void add(int i, T v) {
    if (sz(rui) - 1 != i)
      ole();
    operator+=(v);
  }
  T back() { return rui.back(); }
  ll size() { return rui.size(); }
  auto begin() { return rui.begin(); }
  auto end() { return rui.end(); }
};
template <class T> string deb_tos(const ruiC<T> &a) { return deb_tos(a.rui); }
template <class T> ostream &operator<<(ostream &os, ruiC<T> a) {
  fora(v, a.rui) { os << v << " "; }
  return os;
}
template <class T> vector<T> ruiv(const vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i];
  return ret;
}
template <class T> ruiC<T> ruic(const vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
template <class T> ruiC<T> ruic() { return ruiC<T>(); }
// imoは0-indexed
// ruiは1-indexed
template <class T> vector<T> imo(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}

// #define use_rui //_imo _ruic _ruiv
#ifdef use_rui
// kと同じものの数
template <class T, class U> vi imo(const vector<T> &a, U k) {
  vi equ(sz(a));
  rep(i, sz(a)) { equ[i] = a[i] == k; }
  return imo(equ);
}
template <class T> vector<T> imox(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] ^= ret[i];
  return ret;
}
// 漸化的に最小を持つ
template <class T> vector<T> imi(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chmi(ret[i + 1], ret[i]);
  return ret;
}
template <class T> vector<T> ima(const vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) chma(ret[i + 1], ret[i]);
  return ret;
}
template <class T> vector<T> rimi(const vector<T> &v) {
  vector<T> ret = v;
  rer(i, sz(ret) - 1, 1) chmi(ret[i - 1], ret[i]);
  return ret;
}
template <class T> vector<T> rima(const vector<T> &v) {
  vector<T> ret = v;
  rer(i, sz(ret) - 1, 1) chma(ret[i - 1], ret[i]);
  return ret;
}
template <class T> struct ruimax {
  template <typename Monoid> struct SegmentTree { /*pairで処理*/
    int sz;
    vector<Monoid> seg;
    const Monoid M1 = mp(MIN<T>(), -1);
    Monoid f(Monoid a, Monoid b) { return max(a, b); }
    void build(vector<T> &a) {
      int n = sz(a);
      sz = 1;
      while (sz < n)
        sz <<= 1;
      seg.assign(2 * sz, M1);
      rep(i, n) { seg[i + sz] = mp(a[i], i); }
      for (int k = sz - 1; k > 0; k--) {
        seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);
      }
    }
    Monoid query(int a, int b) {
      Monoid L = M1, R = M1;
      for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
        if (a & 1)
          L = f(L, seg[a++]);
        if (b & 1)
          R = f(seg[--b], R);
      }
      return f(L, R);
    }
    Monoid operator[](const int &k) const { return seg[k + sz]; }
  };

private:
  vector<T> ve;
  SegmentTree<pair<T, int>> seg;
  vector<T> rv;
  vector<int> ri;
  bool build = false;

public:
  int n;
  ruimax(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    T ma = MIN<T>();
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = -INF<T>();
    ri[0] = -1;
    rep(i, n) {
      if (chma(ma, a[i])) {
        index = i;
      }
      rv[i + 1] = ma;
      ri[i + 1] = index;
    }
  }
  T operator()(int l, int r) {
    if (!(l <= r && 0 <= l && r <= n)) {
      deb(l, r, n);
      assert(0);
    }
    if (l == 0) {
      return rv[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).first;
    }
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  T getv(int l, int r) { return operator()(l, r); }
  T getv(int r = inf) { return getv(0, min(r, n)); };
  int geti(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return ri[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).second;
    }
  }
  int geti(int r = inf) { return geti(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};
template <class T> struct ruimin {
  template <typename Monoid> struct SegmentTree { /*pairで処理*/
    int sz;
    vector<Monoid> seg;
    const Monoid M1 = mp(MAX<T>(), -1);
    Monoid f(Monoid a, Monoid b) { return min(a, b); }
    void build(vector<T> &a) {
      int n = sz(a);
      sz = 1;
      while (sz < n)
        sz <<= 1;
      seg.assign(2 * sz, M1);
      rep(i, n) { seg[i + sz] = mp(a[i], i); }
      for (int k = sz - 1; k > 0; k--) {
        seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);
      }
    }
    Monoid query(int a, int b) {
      Monoid L = M1, R = M1;
      for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
        if (a & 1)
          L = f(L, seg[a++]);
        if (b & 1)
          R = f(seg[--b], R);
      }
      return f(L, R);
    }
    Monoid operator[](const int &k) const { return seg[k + sz]; }
  };

private:
  vector<T> ve;
  SegmentTree<pair<T, int>> seg;
  vector<T> rv;
  vector<int> ri;
  bool build = false;
  int n;

public:
  ruimin(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    T mi = MAX<T>();
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = INF<T>();
    ri[0] = -1;
    rep(i, n) {
      if (chmi(mi, a[i])) {
        index = i;
      }
      rv[i + 1] = mi;
      ri[i + 1] = index;
    }
  }
  T operator()(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return rv[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).first;
    }
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  T getv(int l, int r) { return operator()(l, r); }
  T getv(int r = inf) { return getv(0, min(r, n)); };
  int geti(int l, int r) {
    {
      assert(l <= r && 0 <= l && r <= n);
      if (l == 0) {
        return ri[r];
      } else {
        if (!build)
          seg.build(ve), build = true;
        return seg.query(l, r).second;
      }
    }
    assert(l <= r && 0 <= l && r <= n);
    if (l == 0) {
      return ri[r];
    } else {
      if (!build)
        seg.build(ve), build = true;
      return seg.query(l, r).second;
    }
  }
  int geti(int r = inf) { return geti(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
}; /*@formatter:off*/

vvi() ruib(vi &a) {
  vvi(res, 61, sz(a) + 1);
  rep(k, 61) {
    rep(i, sz(a)) { res[k][i + 1] = res[k][i] + ((a[i] >> k) & 1); }
  }
  return res;
}
vector<ruiC<int>> ruibc(vi &a) {
  vector<ruiC<int>> ret(61);
  vvi(res, 61, sz(a));
  rep(k, 61) {
    rep(i, sz(a)) { res[k][i] = (a[i] >> k) & 1; }
    ret[k] = ruic(res[k]);
  }
  return ret;
}
// kと同じものの数
template <class T, class U> vi ruiv(T &a, U k) {
  vi ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + (a[i] == k);
  return ret;
}
template <class T, class U> ruiC<ll> ruic(T &a, U k) {
  vi ret = ruiv(a, k);
  return ruiC<ll>(ret);
}
template <class T> struct ruiC2 {
  int H;
  vector<ruiC<T>> rui;
  ruiC<T> dummy; // 変なのをよばれたときはこれを返す//todo
  ruiC2(const vector<vector<T>> &ru) : rui(sz(ru)), H(sz(ru)) {
    for (int h = 0; h < H; h++) {
      if (sz(ru[h]) == 0)
        continue;
      if (sz(dummy) == 1)
        dummy = ruic(vector<T>(sz(ru[h])));
      rui[h] = ruic(ru[h]);
    }
  }
  // WについてHを返す
  vector<T> operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruic ";
      deb(l, r);
      assert(0);
    }
    vector<T> res(H);
    for (int h = 0; h < H; h++)
      res[h] = rui[h](l, r);
    return res;
  }
  // HについてWを返す
  ruiC<T> &operator[](ll h) {
#ifdef _DEBUG
    if (h >= H) {
      message += "warning ruiC h >= H";
    }
#endif
    if (h >= H || sz(rui[h]) == 1)
      return dummy;
    else
      return rui[h];
  }
  /*@formatter:off*/
  //    vector<T> operator()(int r) { return operator()(0, r); }
  /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
  /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
  //    T operator[](ll i) { return rui[i + 1] - rui[i]; }
  /*0から順に追加される必要がある*/
  //    T back() { return rui.back(); }
  //    ll size() { return rui.size(); }
  //    auto begin(){return rui.begin();}
  //    auto end(){return rui.end();}
};

template <class T, class U> ruiC<ll> ruicou(vector<T> &a, U b) {
  vi cou(sz(a));
  rep(i, sz(a)) { cou[i] = a[i] == b; }
  return ruic(cou);
}

// メモリは形式によらず(26*N)
//  rui(l,r)でvector(26文字について, l~rのcの個数)
//  rui[h] ruic()を返す
//  添え字は'a', 'A'のまま扱う
//  (予め-='a','A'されているものが渡されたらそれに従う)
template <typename Iterable, class is_Iterable = typename Iterable::value_type>
ruiC2<ll> ruicou(const Iterable &a) {
  int H = max(a) + 1;
  vvi(cou, H);
  rep(i, sz(a)) {
    if (sz(cou[a[i]]) == 0)
      cou[a[i]].resize(sz(a));
    cou[a[i]][i] = 1;
  }
  return ruiC2<ll>(cou);
}

/*@formatter:off*/
// h query
template <class T> vector<T> imoh(vector<vector<T>> &v, int w) {
  vector<T> ret(sz(v));
  rep(h, sz(ret)) { ret[h] = v[h][w]; }
  rep(i, sz(ret) - 1) { ret[i + 1] += ret[i]; }
  return ret;
}
template <class T> vector<T> ruih(vector<vector<T>> &v, int w) {
  vector<T> ret(sz(v) + 1);
  rep(h, sz(v)) { ret[h + 1] = v[h][w]; }
  rep(i, sz(v)) { ret[i + 1] += ret[i]; }
  return ret;
}
template <class T> ruiC<T> ruihc(vector<vector<T>> &a, int w) {
  vector<T> ret = ruih(a, w);
  return ruiC<T>(ret);
}

// xor
template <class T> struct ruixC {
  vector<T> rui;
  ruixC(vector<T> &ru) : rui(ru) {}
  T operator()(ll l, ll r) {
    if (l > r) {
      cerr << "ruiXc ";
      deb(l, r);
      assert(0);
    }
    return rui[r] ^ rui[l];
  }
  T operator[](ll i) { return rui[i]; }
  T back() { return rui.back(); }
  ll size() { return rui.size(); }
};
template <class T> vector<T> ruix(vector<T> &a) {
  vector<T> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] ^ a[i];
  return ret;
}
template <class T> ruixC<ll> ruixc(vector<T> &a) {
  vi ret = ruix(a);
  return ruixC<ll>(ret);
}
// 差分を返す(累積を取ると元に戻る)
// 101なら
// 1111を返す
// 元の配列で[l, r)へのxorは
//[l]と[r]へのxorになる https://atcoder.jp/contests/abc155/tasks/abc155_f
vi ruix_diff(vi &A) {
  int N = sz(A);
  assert(N);
  vi res(N + 1);
  res[0] = A[0];
  rep(i, 1, N) { res[i] = A[i - 1] ^ A[i]; }
  res[N] = A[N - 1];
  return res;
}

template <class T> vector<T> ruim(vector<T> &a) {
  vector<T> res(a.size() + 1, 1);
  rep(i, a.size()) res[i + 1] = res[i] * a[i];
  return res;
}
// 漸化的に最小を1indexで持つ
template <class T> vector<T> ruimi(vector<T> &a) {
  ll n = sz(a);
  vector<T> ret(n + 1);
  rep(i, 1, n) {
    ret[i] = a[i - 1];
    chmi(ret[i + 1], ret[i]);
  }
  return ret;
}
// template<class T> T *rrui(vector<T> &a) {
// 右から左にかけての半開区間 (-1 n-1]
template <class T> struct rruiC {
  vector<T> rui;
  int n;
  rruiC(vector<T> &a) : n(sz(a)) {
    rui.resize(n + 1);
    rer(i, n - 1) { rui[i] = rui[i + 1] + a[i]; }
  }
  /*[r l)*/
  T operator()(int r, int l) {
    r++;
    l++;
    assert(l <= r && l >= 0 && r <= n);
    return rui[l] - rui[r];
  }
  T operator()(int l) { return operator()(n - 1, l); }
  T operator[](int i) { return operator()(i); }
};
template <class T> ostream &operator<<(ostream &os, rruiC<T> a) {
  fora(v, a.rui) { os << v << " "; }
  return os;
}
template <class T> string deb_tos(rruiC<T> &a) { return deb_tos(a.rui); }

#define rrui rruic
template <class T> rruiC<T> rruic(vector<T> &a) { return rruiC<T>(a); }

// 掛け算
template <class T> struct ruimulC {
  vector<T> rv;
  int n;
  ruimulC(vector<T> &a) : rv(a), n(sz(a)) {
    rv.resize(n + 1);
    rv[0] = 1;
    rep(i, n) { rv[i + 1] = a[i] * rv[i]; }
  }
  ruimulC() : n(0) {
    rv.resize(n + 1);
    rv[0] = 1;
  }
  void operator+=(T v) {
    rv.push_back(rv.back() * v);
    n++;
  }
  T operator()(int l, int r) {
    assert(l <= r && 0 <= l && r <= n);
    return rv[r] / rv[l];
  }
  T operator()(int r = inf) { return operator()(0, min(r, n)); }
  T operator[](int r) { return operator()(0, r); }
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};
template <class T> ruimulC<T> ruimul(vector<T> &a) { return ruimulC<T>(a); }
template <class T> ruimulC<T> ruimul() {
  vector<T> a;
  return ruimulC<T>(a);
}
template <class T> T *rruim(vector<T> &a) {
  ll len = a.size();
  T *body = (T *)malloc((len + 1) * sizeof(T));
  T *res = body + 1;
  res[len - 1] = 1;
  rer(i, len - 1) res[i - 1] = res[i] * a[i];
  return res;
}
template <class T, class U, class W> T lowerBound(ruiC<T> &a, U v, W banpei) {
  return lowerBound(a.rui, v, banpei);
}
template <class T, class U, class W> T upperBound(ruiC<T> &a, U v, W banpei) {
  return upperBound(a.rui, v, banpei);
}
template <class T, class U, class W> T rlowerBound(ruiC<T> &a, U v, W banpei) {
  return rlowerBound(a.rui, v, banpei);
}
template <class T, class U, class W> T rupperBound(ruiC<T> &a, U v, W banpei) {
  return rupperBound(a.rui, v, banpei);
}
#endif

constexpr bool bget(ll m, ll keta) {
#ifdef _DEBUG
  assert(keta <= 62); // オーバーフロー 1^62までしか扱えない
#endif
  return (m >> keta) & 1;
}
// bget(n)次元
//                           NならN-1まで
vector<vi> bget2(vi &a, int keta_size) {
  vvi(res, keta_size, sz(a));
  rep(k, keta_size) {
    rep(i, sz(a)) { res[k][i] = bget(a[i], k); }
  }
  return res;
}
vi bget1(vi &a, int keta) {
  vi res(sz(a));
  rep(i, sz(a)) { res[i] = bget(a[i], keta); }
  return res;
}
#if __cplusplus >= 201703L
ll bget(ll m, ll keta, ll sinsuu) {
  m /= pow<ll>(sinsuu, keta);
  return m % sinsuu;
}
#else
ll bget(ll m, ll keta, ll sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
#endif
constexpr ll bit(ll n) {
#ifdef _DEBUG
  assert(n <= 62); // オーバーフロー 1^62までしか扱えない
#endif
  return (1LL << (n));
}
#if __cplusplus >= 201703L
ll bit(ll n, ll sinsuu) { return pow<ll>(sinsuu, n); }
#else
ll bit(ll n, ll sinsuu) { return (ll)pow(sinsuu, n); }
#endif
ll mask(ll n) { return (1ll << n) - 1; }
// aをbitに置きなおす
//{0, 2} -> 101
ll bit(const vi &a) {
  int m = 0;
  for (auto &&v : a)
    m |= bit(v);
  return m;
}

//{1, 1, 0} -> 011
// bitsetに置き換える感覚 i が立っていたら i bit目を立てる
ll bit_bool(vi &a) {
  int m = 0;
  rep(i, sz(a)) if (a[i]) m |= bit(i);
  return m;
}
#define bcou __builtin_popcountll
// 最下位ビット
ll lbit(ll n) {
  assert(n);
  return n & -n;
}
ll lbiti(ll n) {
  assert(n);
  return log2(n & -n);
}
// 最上位ビット
ll hbit(ll n) {
  assert(n);
  n |= (n >> 1);
  n |= (n >> 2);
  n |= (n >> 4);
  n |= (n >> 8);
  n |= (n >> 16);
  n |= (n >> 32);
  return n - (n >> 1);
}
ll hbiti(ll n) {
  assert(n);
  return log2(hbit(n));
}
// ll hbitk(ll n) {    ll k = 0;    rer(i, 5) {        ll a = k + (1ll << i); ll
// b = 1ll << a;        if (b <= n)k += 1ll << i;    }    return k;}
// 初期化は0を渡す
ll nextComb(ll &mask, ll n, ll r) {
  if (!mask)
    return mask = (1LL << r) - 1;
  ll x = mask & -mask; /*最下位の1*/
  ll y = mask + x;     /*連続した下の1を繰り上がらせる*/
  ll res = ((mask & ~y) / x >> 1) | y;
  if (bget(res, n))
    return mask = 0;
  else
    return mask = res;
}
// n桁以下でビットがr個立っているもののvectorを返す
vi bitCombList(ll n, ll r) {
  vi res;
  ll m = 0;
  while (nextComb(m, n, r)) {
    res.push_back(m);
  }
  return res;
}
/*over*/ #define forbit1_2(i, mas)                                             \\
  for (int forbitj = !mas ? 0 : lbit(mas), forbitm = mas,                      \\
           i = !mas ? 0 : log2(forbitj);                                       \\
       forbitm; forbitm = forbitm ^ forbitj,                                   \\
           forbitj = !forbitm ? 1 : lbit(forbitm), i = log2(forbitj))
/*over*/ #define forbit1_3(i, N, mas)                                          \\
  for (int forbitj = !mas ? 0 : lbit(mas), forbitm = mas,                      \\
           i = !mas ? 0 : log2(forbitj);                                       \\
       forbitm && i < N; forbitm = forbitm ^ forbitj,                          \\
           forbitj = !forbitm ? 1 : lbit(forbitm), i = log2(forbitj))
// masの立ってるindexを見る
//               i, [N], mas
#define forbit1(...) over3(__VA_ARGS__, forbit1_3, forbit1_2)(__VA_ARGS__)
// masが立っていないindexを見る
//               i, N, mas
#define forbit0(i, N, mas) forbit1(i, (mask(N) & (~(mas))))

// forsubをスニペットして使う
// Mの部分集合(0,M含む)を見る 3^sz(S)個ある
#define forsub_all(m, M) for (int m = M; m != -1; m = m == 0 ? -1 : (m - 1) & M)
// BASE進数
template <size_t BASE> class base_num {
  int v;

public:
  base_num(int v = 0) : v(v){};
  int operator[](int i) { return bget(v, i, BASE); }
  void operator++() { v++; }
  void operator++(signed) { v++; }
  operator int() { return v; }
};
#define base3(mas, lim, BASE) for (base_num<BASE> mas; mas < lim; mas++)
#define base2(mas, lim) base3(mas, lim, 2)
#define base(...) over3(__VA_ARGS__, base3, base2, base1)(__VA_ARGS__)

// aにある物をtrueとする
vb bool_(vi a, int n) {
  vb ret(max(max(a) + 1, n));
  rep(i, sz(a)) ret[a[i]] = true;
  return ret;
}

char itoal(ll i) { return 'a' + i; }
char itoaL(ll i) { return 'A' + i; }
ll altoi(char c) {
  if ('A' <= c && c <= 'Z')
    return c - 'A';
  return c - 'a';
}
ll ctoi(char c) { return c - '0'; }
char itoc(ll i) { return i + '0'; }
ll vtoi(vi &v) {
  ll res = 0;
  if (sz(v) > 18) {
    debugline("vtoi");
    deb(sz(v));
    ole();
  }
  rep(i, sz(v)) {
    res *= 10;
    res += v[i];
  }
  return res;
}
vi itov(ll i) {
  vi res;
  while (i) {
    res.push_back(i % 10);
    i /= 10;
  }
  res = rev(res);
  return res;
}
vi stov(string &a) {
  ll n = sz(a);
  vi ret(n);
  rep(i, n) { ret[i] = a[i] - '0'; }
  return ret;
}
// 基準を満たさないものは0になる
vi stov(string &a, char one) {
  ll n = sz(a);
  vi ret(n);
  rep(i, n) ret[i] = a[i] == one;
  return ret;
}
vector<vector<ll>> ctoi(vector<vector<char>> s, char c) {
  ll n = sz(s), m = sz(s[0]);
  vector<vector<ll>> res(n, vector<ll>(m));
  rep(i, n) rep(j, m) res[i][j] = s[i][j] == c;
  return res;
}
// #define use_compress
//[i] := vを返す
// aは0~n-1で置き換えられる
vi compress(vi &a) {
  vi b;
  ll len = a.size();
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
#ifdef use_compress
// ind[i] := i番目に小さい数
// map[v] := vは何番目に小さいか
vi compress(vi &a, umapi &map) {
  vi b;
  ll len = a.size();
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    ll v = a[i];
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
    map[v] = a[i];
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vi &a, vi &r) {
  vi b;
  ll len = a.size();
  fora(v, a) { b.push_back(v); }
  fora(v, r) { b.push_back(v); }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i)
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  for (ll i = 0; i < sz(r); ++i)
    r[i] = lower_bound(ALL(b), r[i]) - b.begin();
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vi &a, vi &r, vi &s) {
  vi b;
  ll len = a.size();
  fora(v, a) { b.push_back(v); }
  fora(v, r) { b.push_back(v); }
  fora(v, s) { b.push_back(v); }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i)
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  for (ll i = 0; i < sz(r); ++i)
    r[i] = lower_bound(ALL(b), r[i]) - b.begin();
  for (ll i = 0; i < sz(s); ++i)
    r[i] = lower_bound(ALL(b), s[i]) - b.begin();
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vector<vi> &a) {
  vi b;
  fora(vv, a) {
    fora(v, vv) { b.push_back(v); }
  }
  sort(b);
  unique(b);
  fora(vv, a) {
    fora(v, vv) { v = lower_bound(ALL(b), v) - b.begin(); }
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
vi compress(vector<vector<vi>> &a) {
  vi b;
  fora(vvv, a) {
    fora(vv, vvv) {
      fora(v, vv) { b.push_back(v); }
    }
  }
  sort(b);
  unique(b);
  fora(vvv, a) {
    fora(vv, vvv) {
      fora(v, vv) { v = lower_bound(ALL(b), v) - b.begin(); }
    }
  }
  ll blen = sz(b);
  vi ret(blen);
  rep(i, blen) { ret[i] = b[i]; }
  return ret;
}
void compress(ll a[], ll len) {
  vi b;
  for (ll i = 0; i < len; ++i) {
    b.push_back(a[i]);
  }
  sort(b);
  unique(b);
  for (ll i = 0; i < len; ++i) {
    a[i] = lower_bound(ALL(b), a[i]) - b.begin();
  }
}
#endif
// 要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(ALL(a), v))
#define lowerIndex(a, v) (lower_bound(ALL(a), v) - a.begin())
#define upperIndex(a, v) (upper_bound(ALL(a), v) - a.begin())
#define rlowerIndex(a, v) (upper_bound(ALL(a), v) - a.begin() - 1)
#define rupperIndex(a, v) (lower_bound(ALL(a), v) - a.begin() - 1)

template <class T, class U, class W> T lowerBound(vector<T> &a, U v, W banpei) {
  auto it = lower_bound(a.begin(), a.end(), v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(vector<T> &a, U v, W banpei) {
  auto it = upper_bound(a.begin(), a.end(), v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W>
T rlowerBound(vector<T> &a, U v, W banpei) {
  auto it = upper_bound(a.begin(), a.end(), v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W>
T rupperBound(vector<T> &a, U v, W banpei) {
  auto it = lower_bound(a.begin(), a.end(), v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
// todo 消せないか
template <class T, class U, class W> T lowerBound(set<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(set<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T rlowerBound(set<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W> T rupperBound(set<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}

template <class T, class U, class W> T lowerBound(mset<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T upperBound(mset<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.end())
    return banpei;
  else
    return *it;
}
template <class T, class U, class W> T rlowerBound(mset<T> &a, U v, W banpei) {
  auto it = a.upper_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
template <class T, class U, class W> T rupperBound(mset<T> &a, U v, W banpei) {
  auto it = a.lower_bound(v);
  if (it == a.begin())
    return banpei;
  else {
    return *(--it);
  }
}
#define next2(a) next(next(a))
#define prev2(a) prev(prev(a))

// 狭義の単調増加列 長さを返す
template <class T> int lis(vector<T> &a) {
  int n = sz(a);
  vi tail(n + 1, MAX<T>());
  rep(i, n) {
    int id = lowerIndex(tail, a[i]); /**/
    tail[id] = a[i];
  }
  return lowerIndex(tail, MAX<T>());
}
template <class T> int lis_eq(vector<T> &a) {
  int n = sz(a);
  vi tail(n + 1, MAX<T>());
  rep(i, n) {
    int id = upperIndex(tail, a[i]); /**/
    tail[id] = a[i];
  }
  return lowerIndex(tail, MAX<T>());
}

// iteratorを返す
// valueが1以上の物を返す 0は見つけ次第削除
// vを減らす場合 (*it).se--でいい
template <class T, class U, class V> auto lower_map(map<T, U> &m, V k) {
  auto ret = m.lower_bound(k);
  while (ret != m.end() && (*ret).second == 0) {
    ret = m.erase(ret);
  }
  return ret;
}
template <class T, class U, class V> auto upper_map(map<T, U> &m, V k) {
  auto ret = m.upper_bound(k);
  while (ret != m.end() && (*ret).second == 0) {
    ret = m.erase(ret);
  }
  return ret;
}
// 存在しなければエラー
template <class T, class U, class V> auto rlower_map(map<T, U> &m, V k) {
  auto ret = upper_map(m, k);
  assert(ret != m.begin());
  ret--;
  while (1) {
    if ((*ret).second != 0)
      break;
    assert(ret != m.begin());
    auto next = ret;
    --next;
    m.erase(ret);
    ret = next;
  }
  return ret;
}
template <class T, class U, class V> auto rupper_map(map<T, U> &m, V k) {
  auto ret = lower_map(m, k);
  assert(ret != m.begin());
  ret--;
  while (1) {
    if ((*ret).second != 0)
      break;
    assert(ret != m.begin());
    auto next = ret;
    --next;
    m.erase(ret);
    ret = next;
  }
  return ret;
}

template <class... T> void fin(T... s) {
  out(s...);
  exit(0);
}

// 便利 数学 math
// sub ⊂ top
bool subset(int sub, int top) { return (sub & top) == sub; }
//-180 ~ 180 degree
double atand(double h, double w) { return atan2(h, w) / PI * 180; }
//% -mの場合、最小の正の数を返す
ll mod(ll a, ll m) {
  if (m < 0)
    m *= -1;
  return (a % m + m) % m;
}
// ll pow(ll a) { return a * a; };
template <class T> T fact(int v) {
  static vector<T> fact(2, 1);
  if (sz(fact) <= v) {
    rep(i, sz(fact), v + 1) { fact.emplace_back(fact.back() * i); }
  }
  return fact[v];
}

ll comi(ll n, ll r) {
  assert(n < 100);
  static vvi(pas, 100, 100);
  if (pas[0][0])
    return pas[n][r];
  pas[0][0] = 1;
  rep(i, 1, 100) {
    pas[i][0] = 1;
    rep(j, 1, i + 1) pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];
  }
  return pas[n][r];
}
// 二項係数の偶奇を返す
int com_mod2(int n, int r) { return n == (r | (n - r)); }
double comd2(ll n, ll r) {
  static vvd(comb, 2020, 2020);
  if (comb[0][0] == 0) {
    comb[0][0] = 1;
    rep(i, 2000) {
      comb[i + 1][0] = 1;
      rep(j, 1, i + 2) { comb[i + 1][j] = comb[i][j] + comb[i][j - 1]; }
    }
  }
  return comb[n][r];
}
double comd(int n, int r) {
  if (r < 0 || r > n)
    return 0;
  if (n < 2020)
    return comd2(n, r);
  static vd fact(2, 1);
  if (sz(fact) <= n) {
    rep(i, sz(fact), n + 1) { fact.push_back(fact.back() * i); }
  }
  return fact[n] / fact[n - r] / fact[r];
}

#define gcd my_gcd
template <class T, class U, class V = typename max_type<T, U>::type>
V gcd(T a_, U b_) {
  V a = a_;
  V b = b_;
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
template <class T> T gcd(const vector<T> &b) {
  T res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
#define lcm my_lcm
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll lcm(vi a) {
  ll res = a[0];
  rep(i, 1, sz(a)) res = lcm(a[i], res);
  return res;
}
ll ceil(ll a, ll b) {
  if (b == 0) {
    debugline("ceil");
    deb(a, b);
    ole();
    return -1;
  } else if (a < 0) {
    return 0;
  } else {
    return (a + b - 1) / b;
  }
}
#define hypot my_hypot
double hypot(double dx, double dy) { return std::sqrt(dx * dx + dy * dy); }

ll sig0(int t) { return t <= 0 ? 0 : ((1 + t) * t) >> 1; }
bint sig0(bint t) { return t <= 0 ? 0 : ((1 + t) * t) >> 1; }
// ll sig(ll s, ll t) { return ((s + t) * (t - s + 1)) >> 1; }
ll sig(ll s, ll t) {
  if (s > t)
    swap(s, t);
  return sig0(t - s) + s * (t - s + 1);
}

#define tousa_i tosa_i
#define lower_tousa_i lower_tosa_i
#define upper_tousa upper_tosa
#define upper_tousa_i upper_tosa_i
ll tosa_i(ll st, ll ad, ll v) {
  assert(((v - st) % ad) == 0);
  return (v - st) / ad;
}
ll tosa_s(ll st, ll ad, ll len) { return st * len + sig0(len - 1) * ad; }

// ax + r (x は非負整数) で表せる整数のうち、v 以上となる最小の整数
ll lower_tosa(ll st, ll ad, ll v) {
  if (st >= v)
    return st;
  return (v - st + ad - 1) / ad * ad + st;
}
// 第何項か
ll lower_tosa_i(ll st, ll ad, ll v) {
  if (st >= v)
    return 0;
  return (v - st + ad - 1) / ad;
}
ll upper_tosa(ll st, ll ad, ll v) { return lower_tosa(st, ad, v + 1); }
ll upper_tosa_i(ll st, ll ad, ll v) { return lower_tosa_i(st, ad, v + 1); }

// b * res <= aを満たす [l, r)を返す div
P drange_ika(int a, int b) {
  P null_p = mp(linf, linf);
  if (b == 0) {
    if (a >= 0) {
      return mp(-linf, linf + 1) /*全て*/;
    } else {
      return null_p /*無い*/;
    }
  } else {
    if (a >= 0) {
      if (b > 0) {
        return mp(-linf, a / b + 1);
      } else {
        return mp(-(a / -b), linf + 1);
      }
    } else {
      if (b > 0) {
        return mp(-linf, -ceil(-a, b) + 1);
      } else {
        return mp(ceil(-a, -b), linf + 1);
      }
    }
  }
}

// v * v >= aとなる最小のvを返す
ll sqrt(ll a) {
  if (a < 0) {
    debugline("sqrt");
    deb(a);
    ole();
  }
  ll res = (ll)std::sqrt(a);
  while (res * res < a)
    ++res;
  return res;
}
double log(double e, double x) { return log(x) / log(e); }

/*@formatter:off*/

// 機能拡張
#define dtie(a, b)                                                             \\
  int a, b;                                                                    \\
  tie(a, b)
template <class T, class U> string to_string(T a, U b) {
  string res = "";
  res += a;
  res += b;
  return res;
}
template <class T, class U, class V> string to_string(T a, U b, V c) {
  string res = "";
  res += a;
  res += b;
  res += c;
  return res;
}
template <class T, class U, class V, class W>
string to_string(T a, U b, V c, W d) {
  string res = "";
  res += a;
  res += b;
  res += c;
  res += d;
  return res;
}
template <class T, class U, class V, class W, class X>
string to_string(T a, U b, V c, W d, X e) {
  string res = "";
  res += a;
  res += b;
  res += c;
  res += d;
  res += e;
  return res;
}
// todo stringもセットで
// find_ifはsubrを使う
string sub(string &A, int l = -1, int r = -1) {
  set_lr12(l, r, sz(A));
  assert(0 <= l && l <= r && r <= sz(A));
  return A.substr(l, r - l);
}
template <class T> vector<T> sub(const vector<T> &A, int l, int r) {
  assert(0 <= l && l <= r && r <= sz(A));
  vector<T> ret(r - l);
  std::copy(A.begin() + l, A.begin() + r, ret.begin());
  return ret;
}
template <class T> vector<T> sub(const vector<T> &A, int r) {
  return sub(A, 0, r);
}
#if __cplusplus >= 201703L
template <class I, require_t(has_value_type<I>::value)>
I subn(const I &A, int l, int len) {
  return sub(A, l, l + len);
}
#endif
template <class T, class F>
// sub2で呼ぶ
vector<T> sub(const vector<vector<T>> &A, int h, int w, int ah, int aw, F f) {
  vector<T> res;
  while (0 <= h && h < sz(A) && 0 <= w && w < sz(A[h]) && f(A[h][w])) {
    res.emplace_back(A[h][w]);
    h += ah;
    w += aw;
  }
  return res;
}
template <class T>
vector<T> sub(const vector<vector<T>> &A, int h, int w, int ah, int aw) {
  return sub(A, h, w, ah, aw, [&](T v) { return true; });
}
// range_nowを返す(find_ifでしか使われない)
#if __cplusplus >= 201703L
template <class T> auto subr(const vector<T> &A, int l) {
  return range_now(vector<T>(A.begin() + l, A.end()), l);
}
#endif
#define sub25(A, h, w, ah, aw) sub(A, h, w, ah, aw)
#define sub26(A, h, w, ah, aw, siki_r)                                         \\
  sub(A, h, w, ah, aw, [&](auto v) { return v siki_r; })
#define sub27(A, h, w, ah, aw, v, siki)                                        \\
  sub(A, h, w, ah, aw, [&](auto v) { return siki; })
#define sub2(...) over7(__VA_ARGS__, sub27, sub26, sub25)(__VA_ARGS__)

constexpr int bsetlen = 50001;
// constexpr int bsetlen = 5050;
#define bset bitset<bsetlen>
bool operator<(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] < b[i])
      return true;
    if (a[i] > b[i])
      return false;
  }
  return false;
}
bool operator>(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] > b[i])
      return true;
    if (a[i] < b[i])
      return false;
  }
  return false;
}
bool operator<=(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] < b[i])
      return true;
    if (a[i] > b[i])
      return false;
  }
  return true;
}
bool operator>=(bitset<bsetlen> &a, bitset<bsetlen> &b) {
  rer(i, bsetlen - 1) {
    if (a[i] > b[i])
      return true;
    if (a[i] < b[i])
      return false;
  }
  return true;
}
string operator~(string &a) {
  string res = a;
  for (auto &&c : res) {
    if (c == '0')
      c = '1';
    else if (c == '1')
      c = '0';
    else {
      cerr << "cant ~" << a << "must bit" << endl;
      exit(0);
    }
  }
  return res;
}
ostream &operator<<(ostream &os, bset &a) {
  bitset<10> b;
  vi list;
  rep(i, bsetlen) {
    if (a[i])
      list.push_back(i), b[i] = 1;
  }
  os << b << ", " << list;
  return os;
}
int hbiti(bset &a) {
  rer(i, bsetlen) {
    if (a[i])
      return i;
  }
  return -1;
}
#define hk(a, b, c) (a <= b && b < c)

// O(N/64)
bset nap(bset &a, int v) {
  bset r = a | a << v;
  return r;
}
bset nap(bset &a, bset &v) {
  bset r = a;
  rep(i, bsetlen) {
    if (v[i])
      r |= a << i;
  }
  return r;
}

template <class T> int count(set<T> &S, T l, T r) {
  assert(l < r);
  auto it = S.lower_bound(l);
  return it != S.end() && (*it) < r;
}

// template<class T> void seth(vector<vector<T>> &S, int w, vector<T> &v)
// {assert(sz(S) == sz(v));assert(w < sz(S[0]));rep(h, sz(S)) { S[h][w] = v[h];
// }}
template <class T> vector<T> geth(vector<vector<T>> &S, int w) {
  assert(w < sz(S[0]));
  vector<T> ret(sz(S));
  rep(h, sz(S)) { ret[h] = S[h][w]; }
  return ret;
}
// vector<bool>[i]は参照を返さないため、こうしないとvb[i] |=
// trueがコンパイルエラー
vb::reference operator|=(vb::reference a, bool b) { return a = a | b; }
vb::reference operator&=(vb::reference a, bool b) { return a = a & b; }
template <class T, class U>
void operator+=(pair<T, U> &a, const pair<T, U> &b) {
  a.fi += b.fi;
  a.se += b.se;
}
template <class T, class U>
pair<T, U> operator+(const pair<T, U> &a, const pair<T, U> &b) {
  return pair<T, U>(a.fi + b.fi, a.se + b.se);
}
template <class T, class U> pair<T, U> operator*(const pair<T, U> &a, int b) {
  return pair<T, U>(a.fi * b, a.se * b);
}
template <class T, class U>
pair<T, U> operator-(const pair<T, U> &a, const pair<T, U> &b) {
  return pair<T, U>(a.fi - b.fi, a.se - b.se);
}
template <class T, class U> pair<T, U> operator-(const pair<T, U> &a) {
  return pair<T, U>(-a.first, -a.second);
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rhs) {
#ifdef _DEBUG
  static bool was = false;
  if (!was)
    message += "str += 65  is  'A'  not \\"65\\" ";
  was = true;
#endif
  return lhs + (char)rhs;
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const int lhs, const basic_string<CharT, Traits, Alloc> &rhs) {
#ifdef _DEBUG
  static bool was = false;
  if (!was)
    message += "str += 65  is  'A'  not \\"65\\" ";
  was = true;
#endif
  return (char)lhs + rhs;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  lhs = lhs + rv;
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  const int rv2 = rv;
  return lhs + rv2;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  const int v = rv;
  lhs += v;
}
template <typename CharT, typename Traits, typename Alloc>
void operator*=(basic_string<CharT, Traits, Alloc> &s, int num) {
  auto bek = s;
  s = "";
  for (; num; num >>= 1) {
    if (num & 1) {
      s += bek;
    }
    bek += bek;
  }
}
template <class T, class U> void operator+=(queue<T> &a, const U &v) {
  a.push(v);
}
template <class T, class U> void operator+=(deque<T> &a, const U &v) {
  a.push_back(v);
}
template <class T>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, vector<T> &v) {
  fora(d, v) { a.push(d); }
  return a;
}
template <class T, class U>
priority_queue<T, vector<T>, greater<T>> &
operator+=(priority_queue<T, vector<T>, greater<T>> &a, const U &v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, const U &v) {
  a.push(v);
  return a;
}
template <class T> set<T> &operator+=(set<T> &a, const vector<T> &v) {
  fora(d, v) { a.insert(d); }
  return a;
}
template <class T, class U> auto operator+=(set<T> &a, const U &v) {
  return a.insert(v);
}
template <class T, class U> auto operator-=(set<T> &a, const U &v) {
  return a.erase(v);
}
template <class T, class U> auto operator+=(mset<T> &a, const U &v) {
  return a.insert(v);
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, const U &v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, const U &v) {
  a.push_back(v);
  return a;
}
template <class T, class U> vector<T> operator+(U v, const vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(const vector<T> &a, const vector<T> &b) {
  vector<T> ret;
  ret = a;
  fora(v, b) { ret += v; }
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, const vector<T> &b) {
  rep(i, sz(b)) { /*こうしないとa+=aで両辺が増え続けてバグる*/
    a.push_back(b[i]);
  }
  return a;
}
template <class T, class U> map<T, U> &operator+=(map<T, U> &a, map<T, U> &b) {
  for (auto &&bv : b) {
    a[bv.first] += bv.second;
  }
  return a;
}
template <class T, class U>
vector<T> operator+(const vector<T> &a, const U &v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> auto operator+=(uset<T> &a, U v) {
  return a.insert(v);
}
template <class T> vector<T> operator%(vector<T> &a, int v) {
  vi ret(sz(a));
  rep(i, sz(a)) { ret[i] = a[i] % v; }
  return ret;
}
template <class T> vector<T> operator%=(vector<T> &a, int v) {
  rep(i, sz(a)) { a[i] %= v; }
  return a;
}
vi operator&(vi &a, vi &b) {
  assert(sz(a) == sz(b));
  vi ret(sz(a));
  rep(i, sz(a)) { ret[i] = min(a[i], b[i]); }
  return ret;
}
template <class T> void operator+=(mset<T> &a, vector<T> &v) {
  for (auto &&u : v)
    a.insert(u);
}
template <class T> void operator+=(set<T> &a, vector<T> &v) {
  for (auto &&u : v)
    a.insert(u);
}
template <class T> void operator+=(vector<T> &a, set<T> &v) {
  for (auto &&u : v)
    a.emplace_back(u);
}
template <class T> void operator+=(vector<T> &a, mset<T> &v) {
  for (auto &&u : v)
    a.emplace_back(u);
}
template <class T> vector<T> &operator-=(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-=");
    deb(a);
    deb(b);
    exit(0);
  }
  rep(i, sz(a)) a[i] -= b[i];
  return a;
}
template <class T> vector<T> operator-(vector<T> &a, vector<T> &b) {
  if (sz(a) != sz(b)) {
    debugline("vector<T> operator-");
    deb(a);
    deb(b);
    ole();
  }
  vector<T> res(sz(a));
  rep(i, sz(a)) res[i] = a[i] - b[i];
  return res;
}
// template<class T, class U> void operator*=(vector<T> &a, U b) {    vector<T>
// ta = a;    rep(b-1){        a+=ta;    }}
template <typename T> void erase(T &v, unsigned ll i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(T &v, unsigned ll s, unsigned ll e) {
  v.erase(v.begin() + s, v.begin() + e);
}
template <class T, class... I> auto erased(T &v, I... arg) {
  auto ret = v;
  erase(ret, arg...);
  return ret;
}
template <typename T> void pop_front(vector<T> &v) { erase(v, 0); }
template <typename T> void entry(vector<T> &v, unsigned ll s, unsigned ll e) {
  erase(v, e, sz(v));
  erase(v, 0, s);
}

template <class T, class U> void erase(map<T, U> &m, ll okl, ll ngr) {
  m.erase(m.lower_bound(okl), m.lower_bound(ngr));
}
template <class T> void erase(set<T> &m, ll okl, ll ngr) {
  m.erase(m.lower_bound(okl), m.lower_bound(ngr));
}
template <typename T> void erasen(vector<T> &v, unsigned ll s, unsigned ll n) {
  v.erase(v.begin() + s, v.begin() + s + n);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned ll i, U t) {
  v.insert(v.begin() + i, t);
}
template <typename T, typename U> void push_front(vector<T> &v, U t) {
  v.insert(v.begin(), t);
}
template <typename T, typename U>
void insert(vector<T> &v, unsigned ll i, vector<T> list) {
  for (auto &&va : list)
    v.insert(v.begin() + i++, va);
}
vector<string> split(const string &a, const char deli) {
  string b = a + deli;
  ll l = 0, r = 0, n = b.size();
  vector<string> res;
  rep(i, n) {
    if (b[i] == deli) {
      r = i;
      if (l < r)
        res.push_back(b.substr(l, r - l));
      l = i + 1;
    }
  }
  return res;
}
vector<string> split(const string &a, const string deli) {
  vector<string> res;
  ll kn = sz(deli);
  std::string::size_type Pos(a.find(deli));
  ll l = 0;
  while (Pos != std::string::npos) {
    if (Pos - l)
      res.push_back(a.substr(l, Pos - l));
    l = Pos + kn;
    Pos = a.find(deli, Pos + kn);
  }
  if (sz(a) - l)
    res.push_back(a.substr(l, sz(a) - l));
  return res;
}
ll stoi(string &s) { return stol(s); }
#define assert_yn(yn_v, v)                                                     \\
  ;                                                                            \\
  assert(yn_v == 0 || yn_v == v);                                              \\
  yn_v = v;
// 不完全な対策、現状はautohotkeyで対応
int yn_v = 0;
void yn(bool a) {
  assert_yn(yn_v, 1);
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void fyn(bool a) {
  assert_yn(yn_v, 1);
  yn(a);
  exit(0);
}

void Yn(bool a) {
  assert_yn(yn_v, 2);
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void fYn(bool a) {
  assert_yn(yn_v, 2);
  Yn(a);
  exit(0);
}

void YN(bool a) {
  assert_yn(yn_v, 3);
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fYN(bool a) {
  assert_yn(yn_v, 3);
  YN(a);
  exit(0);
}
int ab_v = 0;
void fAb(bool a) {
  assert_yn(ab_v, 1);
  if (a)
    cout << "Alice" << endl;
  else
    cout << "Bob";
}
void fAB(bool a) {
  assert_yn(yn_v, 2);
  if (a)
    cout << "ALICE" << endl;
  else
    cout << "BOB";
}
int pos_v = 0;
void Possible(bool a) {
  assert_yn(pos_v, 1);
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}
void POSSIBLE(bool a) {
  assert_yn(pos_v, 2);
  if (a)
    cout << "POSSIBLE" << endl;
  else
    cout << "IMPOSSIBLE" << endl;
  exit(0);
}
void fPossible(bool a) {
  assert_yn(pos_v, 1) Possible(a);
  exit(0);
}
void fPOSSIBLE(bool a) {
  assert_yn(pos_v, 2) POSSIBLE(a);
  exit(0);
}
template <typename T> class fixed_point : T {
public:
  explicit constexpr fixed_point(T &&t) noexcept : T(std::forward<T>(t)) {}
  template <typename... Args>
  constexpr decltype(auto) operator()(Args &&...args) const {
    return T::operator()(*this, std::forward<Args>(args)...);
  }
};
template <typename T>
static inline constexpr decltype(auto) fix(T &&t) noexcept {
  return fixed_point<T>{std::forward<T>(t)};
}

// 未分類
// 0,2,1 1番目と2番目の次元を入れ替える
template <class T>
auto irekae(vector<vector<vector<T>>> &A, int x, int y, int z) {
#define irekae_resize_loop(a, b, c)                                            \\
  resize(res, a, b, c);                                                        \\
  rep(i, a) rep(j, b) rep(k, c)
  vector<vector<vector<T>>> res;
  if (x == 0 && y == 1 && z == 2) {
    res = A;
  } else if (x == 0 && y == 2 && z == 1) {
    irekae_resize_loop(sz(A), sz(A[0][0]), sz(A[0])) {
      res[i][j][k] = A[i][k][j];
    }
  } else if (x == 1 && y == 0 && z == 2) {
    irekae_resize_loop(sz(A[0]), sz(A), sz(A[0][0])) {
      res[i][j][k] = A[j][i][k];
    }
  } else if (x == 1 && y == 2 && z == 0) {
    irekae_resize_loop(sz(A[0]), sz(A[0][0]), sz(A)) {
      res[i][j][k] = A[k][i][j];
    }
  } else if (x == 2 && y == 0 && z == 1) {
    irekae_resize_loop(sz(A[0][0]), sz(A), sz(A[0])) {
      res[i][j][k] = A[j][k][i];
    }
  } else if (x == 2 && y == 1 && z == 0) {
    irekae_resize_loop(sz(A[0][0]), sz(A[0]), sz(A)) {
      res[i][j][k] = A[k][j][i];
    }
  }
  return res;
#undef irekae_resize_loop
}
template <class T> auto irekae(vector<vector<T>> &A, int i = 1, int j = 0) {
  vvt(res, sz(A[0]), sz(A));
  rep(i, sz(A)) {
    rep(j, sz(A[0])) { res[j][i] = A[i][j]; }
  }
  return res;
}
// tou分割する
template <typename Iterable>
vector<Iterable> cut(const Iterable &a, int tou = 2) {
  int N = sz(a);
  vector<Iterable> res(tou);
  int hab = N / tou;
  vi lens(tou, hab);
  rep(i, N % tou) { lens[tou - 1 - i]++; }
  int l = 0;
  rep(i, tou) {
    int len = lens[i];
    int r = l + len;
    res[i].resize(len);
    std::copy(a.begin() + l, a.begin() + r, res[i].begin());
    l = r;
  }
  return res;
}
// 長さn毎に分割する
template <typename Iterable>
vector<Iterable> cut_n(const Iterable &a, int len) {
  int N = sz(a);
  vector<Iterable> res(ceil(N, len));
  vi lens(N / len, len);
  if (N % len)
    lens.push_back(N % len);
  int l = 0;
  rep(i, sz(lens)) {
    int len = lens[i];
    int r = l + len;
    res[i].resize(len);
    std::copy(a.begin() + l, a.begin() + r, res[i].begin());
    l = r;
  }
  return res;
}
// 縦を返す
vi &geth(vvi() & a, int w) {
  static vi ret;
  ret.resize(sz(a));
  rep(i, sz(a)) { ret[i] = a[i][w]; }
  return ret;
}

//@起動時
struct initon {
  initon() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout.setf(ios::fixed);
    cout.precision(16);
    srand((unsigned)clock() + (unsigned)time(NULL));
  };
} initonv;
// #define pre prev
// #define nex next

// gra mll pr
// 上下左右
const string udlr = "udlr";
string UDLR = "UDLR"; // x4と連動 UDLR.find('U') := x4[0]
vc atoz = {'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm',
           'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z'};
vc AtoZ = {'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M',
           'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z'};
// 右、上が正
constexpr ll h4[] = {1, -1, 0, 0};
constexpr ll w4[] = {0, 0, -1, 1};
constexpr ll h8[] = {0, 1, 0, -1, -1, 1, 1, -1};
constexpr ll w8[] = {1, 0, -1, 0, 1, -1, 1, -1};
int mei_inc(int h, int w, int H, int W, int i) {
  while (++i < 4) {
    if (inside(h + h4[i], w + w4[i], H, W))
      return i;
  }
  return i;
}
#define mei(nh, nw, h, w)                                                      \\
  for (int i = mei_inc(h, w, H, W, -1), nh = i < 4 ? h + h4[i] : 0,            \\
           nw = i < 4 ? w + w4[i] : 0;                                         \\
       i < 4; i = mei_inc(h, w, H, W, i), nh = h + h4[i], nw = w + w4[i])
int mei_inc8(int h, int w, int H, int W, int i) {
  while (++i < 8) {
    if (inside(h + h8[i], w + w8[i], H, W))
      return i;
  }
  return i;
}
#define mei8(nh, nw, h, w)                                                     \\
  for (int i = mei_inc8(h, w, H, W, -1), nh = i < 8 ? h + h8[i] : 0,           \\
           nw = i < 8 ? w + w8[i] : 0;                                         \\
       i < 8; i = mei_inc8(h, w, H, W, i), nh = h + h8[i], nw = w + w8[i])
int mei_incv(int h, int w, int H, int W, int i, vp &p) {
  while (++i < sz(p)) {
    if (inside(h + p[i].fi, w + p[i].se, H, W))
      return i;
  }
  return i;
}
#define meiv(nh, nw, h, w, p)                                                  \\
  for (int i = mei_incv(h, w, H, W, -1, p), nh = i < sz(p) ? h + p[i].fi : 0,  \\
           nw = i < sz(p) ? w + p[i].se : 0;                                   \\
       i < sz(p);                                                              \\
       i = mei_incv(h, w, H, W, i, p), nh = h + p[i].fi, nw = w + p[i].se)

// H*Wのグリッドを斜めに分割する
// 右上
vector<vp> naname_list_ne(int H, int W) {
  vector<vp> res(H + W - 1);
  rep(sh, H) {
    int sw = 0;
    res[sh] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh--;
      nw++;
      if (0 <= nh && nw < W) {
        res[sh] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  rep(sw, 1, W) {
    int sh = H - 1;
    res[H + sw - 1] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh--;
      nw++;
      if (0 <= nh && nw < W) {
        res[H + sw - 1] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  return res;
}
// 右下
vector<vp> naname_list_se(int H, int W) {
  vector<vp> res(H + W - 1);
  rep(sh, H) {
    int sw = 0;
    res[sh] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh++;
      nw++;
      if (0 <= nh && nh < H && nw < W) {
        res[sh] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  rep(sw, 1, W) {
    int sh = 0;
    res[H + sw - 1] += mp(sh, sw);
    int nh = sh;
    int nw = sw;
    while (1) {
      nh++;
      nw++;
      if (0 <= nh && nh < H && nw < W) {
        res[H + sw - 1] += mp(nh, nw);
      } else {
        break;
      }
    }
  }
  return res;
}

// グラフ内で #undef getid
// #define getidとしているため、ここを書き直したらgraphも書き直す
#define getid_2(h, w) ((h) * (W) + (w))
#define getid_1(p) ((p).first * W + (p).second)
#define getid(...) over2(__VA_ARGS__, getid_2, getid_1)(__VA_ARGS__)
#define getp(id) mp((id) / W, (id) % W)
#define set_shuffle()                                                          \\
  std::random_device seed_gen;                                                 \\
  std::mt19937 engine(seed_gen())
#define shuffle(a) std::shuffle((a).begin(), (a).end(), engine);
// 1980 開始からtime ms経っていたらtrue
vb bit_bool(int v, int len) {
  assert(bit(len) > v);
  vb ret(len);
  rep(i, len) { ret[i] = bget(v, i); }
  return ret;
}

vi tov(vb &a) {
  vi ret;
  rep(i, sz(a)) {
    if (a[i])
      ret.push_back(i);
  }
  return ret;
}
bool kaibun(const str &S) { return S == rev(S); }
template <class T> vector<T> repeat(const vector<T> &A, int kaisu) {
  vector<T> ret;
  while (kaisu--) {
    ret += A;
  }
  return ret;
}
#define rge range
#define upd update
// S[{s, t, d}]
#define strs slice_str
struct slice_str {
  string S;
  slice_str() {}
  slice_str(const string &S) : S(S) {}
  slice_str(int len, char c) : S(len, c) {}
  auto size() { return S.size(); }
  char &operator[](int p) { return S[p]; }
  string operator[](initializer_list<int> p) {
    if (sz(p) == 1) {
      return S.substr(0, *(p.begin()));
    } else if (sz(p) == 2) {
      int l = *(p.begin());
      int r = *(next(p.begin()));
      return S.substr(l, r - l);
    } else {
      auto it = p.begin();
      int s = *(it++);
      int t = *(it++);
      int d = *(it);
      if (d == -1) {
        int s_ = sz(S) - s - 1;
        int t_ = sz(S) - t - 1;
        return rev(S).substr(s_, t_ - s_);
      } else if (d < 0) {
        t = max(-1ll, t);
        string ret;
        while (s > t) {
          ret += S[s];
          s += d;
        }
        return ret;
      } else {
        t = min(sz(S), t);
        string ret;
        while (s < t) {
          ret += S[s];
          s += d;
        }
        return ret;
      }
    }
  }
  operator string &() { return S; }
  template <class T> void operator+=(const T &a) { S += a; }
  bool operator==(const slice_str &rhs) { return S == rhs.S; }
  auto pop_back() { return S.pop_back(); }
};
ostream &operator<<(ostream &os, const slice_str &a) {
  os << a.S;
  return os;
}
istream &operator>>(istream &iss, slice_str &a) {
  iss >> a.S;
  return iss;
}
template <class T> bool can(const T &v, int i) { return 0 <= i && i < sz(v); }
#if __cplusplus >= 201703L
// template<class T> auto sum(int a, T  v...) {return (v + ... + 0);}
#endif
#define VEC vector

// string.substr()は長さを指定するため、間違えやすい
#define substr my_substr

template <class T> struct my_set : public set<T> {
  template <class... Arg> my_set(Arg... arg) : set<T>(arg...) {}
  T back() { return *(set<T>::rbegin()); }
  T front() { return *(set<T>::begin()); }
};
#define set my_set

#endif /*UNTITLED15_TEMPLATE_H*/

#endif

// † ←template終了
/*@formatter:on*/
// vectorで取れる要素数
// bool=> 1e9 * 8.32
// int => 1e8 * 2.6
// ll  => 1e8 * 1.3
// 3次元以上取るとメモリがヤバい
// static配列を使う
vvc(ba);
ll N, M, H, W;
vi A, B, C;

#ifndef MINT
#define MINT
/*@formatter:off*/
template <typename Mod_t> class modint;
template <typename T> T minv(T a, T m);
template <typename T> T minv(T a);

using Type = std::int_fast64_t;
constexpr signed MOD = 998244353;
//        1e9 + 7;//MOD

using mint = modint<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;

template <typename Mod_t> class modint {
  constexpr static auto mod() { return Mod_t::value; }

public:
  Type value;
  constexpr modint() : value() {}
  constexpr modint(const Type &x) noexcept {
    if (-mod() <= x && x < mod())
      value = static_cast<Type>(x);
    else
      value = static_cast<Type>(x % mod());
    if (value < 0)
      value += mod();
  }
  const Type &operator()() const { return value; }
  template <typename U> explicit operator U() const {
    return static_cast<U>(value);
  }
  constexpr modint operator+(const modint rhs) const noexcept {
    return modint(*this) += rhs;
  }
  constexpr modint operator-(const modint rhs) const noexcept {
    return modint(*this) -= rhs;
  }
  constexpr modint operator*(const modint rhs) const noexcept {
    return modint(*this) *= rhs;
  }
  constexpr modint operator/(const modint rhs) const noexcept {
    return modint(*this) /= rhs;
  }
  constexpr modint &operator+=(const modint rhs) noexcept {
    value += rhs.value;
    if (value >= mod()) {
      value -= mod();
    }
    return *this;
  }
  constexpr modint &operator-=(const modint rhs) noexcept {
    if (value < rhs.value) {
      value += mod();
    }
    value -= rhs.value;
    return *this;
  }
  constexpr modint &operator*=(const modint rhs) noexcept {
    value = value * rhs.value % mod();
    return *this;
  }
  constexpr modint &operator/=(const modint &rhs) noexcept {
    return operator*=(minv(rhs.value));
  }
  modint &operator++() { return *this += 1; }
  modint &operator--() { return *this -= 1; }
  modint operator++(signed) {
    modint result(*this);
    *this += 1;
    return result;
  }
  modint operator--(signed) {
    modint result(*this);
    *this -= 1;
    return result;
  }
  modint operator-() const { return modint(-value); }
  template <typename U>
  friend bool operator==(const modint<U> &lhs, const modint<U> &rhs);
  template <typename U>
  friend bool operator<(const modint<U> &lhs, const modint<U> &rhs);
  template <typename U>
  friend std::istream &operator>>(std::istream &stream, modint<U> &number);
  operator int() { return value; }
};
/*@formatter:off*/
template <typename T>
bool operator==(const modint<T> &lhs, const modint<T> &rhs) {
  return lhs.value == rhs.value;
}
template <typename T, typename U> bool operator==(const modint<T> &lhs, U rhs) {
  return lhs == modint<T>(rhs);
}
template <typename T, typename U> bool operator==(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) == rhs;
}
template <typename T>
bool operator!=(const modint<T> &lhs, const modint<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U> bool operator!=(const modint<T> &lhs, U rhs) {
  return !(lhs == rhs);
}
template <typename T, typename U> bool operator!=(U lhs, const modint<T> &rhs) {
  return !(lhs == rhs);
}
template <typename T>
bool operator<(const modint<T> &lhs, const modint<T> &rhs) {
  return lhs.value < rhs.value;
}
template <typename T>
modint<T> operator+(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T, typename U>
modint<T> operator+(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T, typename U>
modint<T> operator+(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) += rhs;
}
template <typename T>
modint<T> operator-(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T, typename U>
modint<T> operator-(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T, typename U>
modint<T> operator-(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) -= rhs;
}
template <typename T>
modint<T> operator*(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T, typename U>
modint<T> operator*(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T, typename U>
modint<T> operator*(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) *= rhs;
}
template <typename T>
modint<T> operator/(const modint<T> &lhs, const modint<T> &rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator/(const modint<T> &lhs, U rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator/(U lhs, const modint<T> &rhs) {
  return modint<T>(lhs) /= rhs;
}
template <typename T, typename U>
modint<T> operator%(const modint<T> &lhs, U rhs) {
  cerr << "can't  mint % int" << endl;
  assert(0);
  exit(0);
  return 0;
}

#ifdef optional
template <class T>
optional<mint> operator+(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv + opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator-(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator*(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator/(const mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    return mv / opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator+(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv + opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator-(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator*(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv * opt.value();
  else
    return optional<mint>();
}
template <class T>
optional<mint> operator/(const optional<T> &opt, const mint &mv) {
  if (opt.has_value())
    return mv / opt.value();
  else
    return optional<mint>();
}
template <class T> void operator+=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv += opt.value();
}
template <class T> void operator-=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv -= opt.value();
}
template <class T> void operator*=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv *= opt.value();
}
template <class T> void operator/=(mint &mv, const optional<T> &opt) {
  if (opt.has_value())
    mv /= opt.value();
}
template <class T> void operator+=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value += mv;
}
template <class T> void operator-=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value -= mv;
}
template <class T> void operator*=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value *= mv;
}
template <class T> void operator/=(const optional<T> &opt, mint &mv) {
  if (opt.has_value())
    opt.value /= mv;
}
#endif
// constexpr int mint_len = std::min(1400001, MOD);
vector<Type> fac, finv, inv;
vector<Type> p2;
void setmod_resize(int mint_len) {
  //    p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i]
  //    = p2[i - 1] * 2 % MOD;
  int tn = sz(fac);
  fac.resize(mint_len);
  finv.resize(mint_len);
  inv.resize(mint_len);
  inv[1] = fac[0] = finv[0] = 1;
  for (int i = tn; i < mint_len; ++i)
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  for (int i = tn; i < mint_len; ++i)
    fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
}
// 4e6 >= n なら O(r)で計算する
mint com_big(int n, int r) {
  Type fa = finv[r];
  rep(i, r) fa *= n - i, fa %= MOD;
  return mint(fa);
}
mint com(int n, int r) {
  if (r < 0 || r > n)
    return 0; /*nが大きくてrが小さい場合、nを上からr個掛ける*/
  if (n >= sz(fac)) {
    if (n <= 4e6)
      setmod_resize(n + 1000);
    else
      return com_big(n, r);
  }
  return mint(finv[r] * fac[n] % MOD * finv[n - r]);
}
mint pom(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
mint npr(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return mint(fac[n] * finv[n - r]);
}
mint com_grid(int n, int r) {
  assert(r);
  return com(n + r - 2, n - 1);
}
Type nprin(int n, int r) { /*    if (!sz(fac)) com(0, -1);*/
  if (r < 0 || r > n)
    return 0;
  return fac[n] * finv[n - r] % MOD;
}
Type icom(int n, int r) {
  const int NUM_ = 1400001;
  static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];
  if (fac[0] == 0) {
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i <= NUM_; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i <= NUM_; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
  if (r < 0 || r > n)
    return 0;
  return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;
}
#define ncr com
#define ncri icom
// n種類でr個選ぶ
// n個の場所にr個の物を置く
// 負が渡された場合もcomが処理出来る
mint nhr(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return com(n + r - 1, r);
}
// n種類でそれぞれ少なくとも1つは選び、r個揃える
mint nhr1(int n, int r) { return nhr(n, r - n); }
mint hom(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return com(n + r - 1, r);
}
Type nhri(int n, int r) {
  if (n == 0 && r == 0)
    return 1;
  else
    return icom(n + r - 1, r);
}
// グリッドで0-indexedの最短経路 pascal
mint pas(int h, int w) { return com(h + w, w); }
template <typename T> T minv(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
template <typename T> T minv(T a) {
  if (a < sz(fac)) {
    return inv[a];
  }
  T u = 0, v = 1;
  T m = MOD;
  while (a != 0) {
    T t = m / a;
    m -= t * a;
    swap(a, m);
    u -= t * v;
    swap(u, v);
  }
  assert(m == 1);
  return u;
}
// int型 mod m
template <typename T, typename U> mint mpow(const T &a, const U &b) {
  if (b < 0)
    return minv(mpow(a, -b));
  mint x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1)
      res *= x;
    x *= x;
    p >>= 1;
  }
  return res;
}
template <typename T, typename U, typename V>
int mpowi(const T &a, const U &b, const V &m = MOD) {
  return mpow(a, b, m);
}
template <typename T> string to_string(const modint<T> &number) {
  return to_string(number());
}
/*@formatter:off*/
#ifdef _DEBUG
void yuri(const mint &a) {
  stringstream st;
  rep(i, 300) {
    rep(j, 300) {
      if ((mint)i / j == a) {
        st << i << " / " << j;
        i = 2000;
        break;
      }
    }
  }
  string val = st.str();
  if (val != "") {
    deb(val);
    return;
  }
  rep(i, 1000) {
    rep(j, 1000) {
      if ((mint)i / j == a) {
        st << i << " / " << j;
        i = 2000;
        break;
      }
    }
  }
  val = st.str();
  deb(val);
}
#else
#define yuri(...) ;
#endif
template <typename T>
std::ostream &operator<<(std::ostream &stream, const modint<T> &number) {
  stream << number();
#ifdef _DEBUG
//    stream << " -> " << yuri(number);
#endif
  return stream;
}
//@formatter:off
/*@formatter:on*/
template <typename T>
std::istream &operator>>(std::istream &stream, modint<T> &number) {
  typename common_type<typename modint<T>::Type, Type>::type x;
  stream >> x;
  number.value = modint<T>::normalize(x);
  return stream;
}
/*@formatter:off*/
using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define vvm(...)                                                               \\
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(mint, __VA_ARGS__)
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)
string out_m2(mint a) {
  stringstream st;
  st << (int)a;
  rep(i, 300) {
    rep(j, 2, 300) {
      if ((i % j) && (mint)i / j == a) {
        st << "(" << i << "/" << j << ")";
        i = 2000;
        break;
      }
    }
  }
  return st.str();
}
// nhr n個の場所にr個の物を分ける
mint m1 = (mint)1;
mint half = (mint)1 / 2;
struct setmod {
  // comで長さが足りない場合はsetmod_resizeでリサイズする
  int mint_len = 1.4 * 1e6;
  setmod() {
    // p2を使う場合setmod_resizeも変える
    //    p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i)
    //    p2[i] = p2[i - 1] * 2 % MOD;
    fac.resize(mint_len);
    finv.resize(mint_len);
    inv.resize(mint_len);
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i < mint_len; ++i)
      inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i < mint_len; ++i)
      fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
  }
} setmodv;
template <signed S>
using mint_ = modint<std::integral_constant<decay<signed>::type, S>>;
#endif
/*@formatter:on*/
void solve() {
  din(A, B, C, D);

  int H = C - A;
  int W = N;
  if (H <= W) {
    swap(A, B);
    swap(C, D);
  }
  // i番目以降をsから始めた場合の組
  // 累積を取る
  int N = D - B;
  int l = A;
  int r = C;

  vvm(dp, N, 3030);
  rer(s, r, l) { dp[N - 1][s] = s; }
  rer(i, N - 2) {
    mint su = 0;
    rer(s, r, l) {
      su += dp[i + 1][s];
      mint pat = s * su;
      dp[i][s] = pat;
    }
  }
  mint res = 0;
  rep(s, l, r + 1) { res += dp[0][s]; }
  res *= mpow(W, H - W);
  deb(dp);
  deb(H, W);
  out(res);
}

auto my(int A, int B, int C, int D) {
  // i番目以降をsから始めた場合の組
  // 累積を取る
  int N = D - B;
  int l = A;
  int r = C;

  vvm(dp, N, 300);
  rer(s, r, l) { dp[N - 1][s] = s; }
  rer(i, N - 2) {
    mint su = 0;
    rer(s, r, l) {
      su += dp[i + 1][s];
      mint pat = s * su;
      dp[i][s] = pat;
    }
  }
  mint res = 0;
  rep(s, l, r + 1) { res += dp[0][s]; }
  return res;
}

auto sister(int A, int B, int C, int D) {
  int l = A;
  int r = C + 1;
  int N = D - B;
  vi list(N, l);
  mint res = 0;
  rep(mas, pow(r - l, N)) {
    rep(i, N) { list[i] = bget(mas, i, r - l) + l; }
    if (is_sorted(ALL(list))) {
      mint s = 1;
      rep(i, N) { s *= list[i]; }
      res += s;
      //            deb(list);
    }
  }
  return res;
}

signed main() {
  solve();
  return 0;
#ifdef _DEBUG

  bool bad = 0;
  for (ll i = 0, ok = 1; i < k5 && ok; ++i) {
    ll A = rand(1, 200);
    ll B = rand(1, 200);
    ll C = rand(A + 1, 300);
    ll D = rand(B + 1, 300);
    auto myres = my(A, B, C, D);
    auto res = sister(A, B, C, D);
    ok = myres == res;
    if (!ok) {
      deb(A, B, C, D);
      cerr << "AC : " << res << endl;
      cerr << "MY  : " << myres << endl;
      bad = 1;

      break;
    }
  }
  if (!bad) {
    // solveを書き直す
    // solveを呼び出す
  }
  exit(0);
#endif
  return 0;
};