Problem p02782 (C++)

Buggy submission (status: Runtime Error):
// #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 <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;
#define int long long // todo 消したら動かない intの代わりにsignedを使う
#define ll long long
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

/*@formatter:off*/
//[-n, n)にアクセスできる
// また、外部関数resizeに渡せる
// sizeは[-n, n)でnを返す
template <class T> class mvec {
  vector<T> v;
  int n;

public:
  mvec() : n(0), v(0) {}
  mvec(int n) : n(n), v(n * 2) {}
  mvec(int n, T val) : n(n), v(n * 2, val) {}
  auto &operator[](int i) { return v[i + n]; }
  auto size() { return n; }
  void resize(int sn) {
    assert(n == 0);
    n = sn;
    v.resize(sn * 2);
  }
  auto begin() { return v.begin(); }
  auto rbegin() { return v.rbegin(); }
  auto end() { return v.end(); }
  auto rend() { return v.rend(); }
};
//[]でboolは参照を返さないため特殊化が必要
template <> struct mvec<bool> {
  vector<bool> v;
  int n;
  mvec() : n(0), v(0) {}
  mvec(int n) : n(n), v(n * 2) {}
  mvec(int n, bool val) : n(n), v(n * 2, val) {}
  auto operator[](int i) { return v[i + n]; }
  auto size() { return v.size(); }
  void resize(int sn) {
    assert(n == 0);
    n = sn;
    v.resize(sn * 2);
  }
  auto begin() { return v.begin(); }
  auto rbegin() { return v.rbegin(); }
  auto end() { return v.end(); }
  auto rend() { return v.rend(); }
};
template <class T> ostream &operator<<(ostream &os, mvec<T> &a) {
  int spa = 3;
  for (auto &&v : a) {
    spa = max(spa, (int)(to_string(v).size()) + 1);
  }
  int n = (int)a.size();
  os << endl;
  for (int i = -n; i < n; i++) {
    int need = spa - ((int)to_string(i).size());
    if (i == -n) {
      need -= min(need, spa - ((int)to_string(a[i]).size()));
    }
    while (need--) {
      os << " ";
    }
    os << i;
  }
  os << endl;
  int i = -n;
  for (auto &&v : a) {
    int need = spa - ((int)to_string(v).size());
    if (i == -n) {
      need -= min(need, spa - ((int)to_string(i).size()));
    }
    while (need--) {
      os << " ";
    }
    os << v;
    i++;
  }
  return os;
}
#define mv mvec
#define MV mvec
using mvi = mvec<ll>;
using mvb = mvec<bool>;
using mvs = mvec<string>;
using mvd = mvec<double>;
using mvc = mvec<char>;

#define mvvt0(t) mvec<mvec<t>>
#define mvvt1(t, a) mvec<mvec<t>> a
#define mvvt2(t, a, b) mvec<mvec<t>> a(b)
#define mvvt3(t, a, b, c) mvec<mvec<t>> a(b, mvec<t>(c))
#define mvvt4(t, a, b, c, d) mvec<mvec<t>> a(b, mvec<t>(c, d))
#define mvvi(...)                                                              \
  over4(__VA_ARGS__, mvvt4, mvvt3, mvvt2, mvvt1, mvvt0)(ll, __VA_ARGS__)

template <typename T> mvec<T> make_mv(size_t a) { return mvec<T>(a); }
template <typename T, typename... Ts> auto make_mv(size_t a, Ts... ts) {
  return mvec<decltype(make_mv<T>(ts...))>(a, make_mv<T>(ts...));
}
#define mvni(name, ...) auto name = make_mv<ll>(__VA_ARGS__)

#ifdef _DEBUG
string message;
// 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
  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
// 区間削除は出来ない
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>
// gp_hash_tableでcountを使えないようにするため
template <class T, class U> struct my_unordered_map {
  unordered_map<T, U> m;
  my_unordered_map(){};
  auto begin() { return m.begin(); }
  auto end() { return m.end(); }
  auto cbegin() { return m.cbegin(); }
  auto cend() { return m.cend(); }
  template <class V> auto erase(V v) { return m.erase(v); }
  void clear() {
    m.clear();
  } /*countは gp_hash_tableに存在しない*/ /*!= m.end()*/
  template <class V> auto find(V v) { return m.find(v); }
  template <class V> auto &operator[](V n) { return m[n]; }
};
#define unordered_map my_unordered_map
#define umapi unordered_map<ll, ll>
#define umapp unordered_map<P, ll>
#define umapu unordered_map<uint64_t, ll>
#define umapip unordered_map<ll, P>

#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>

#define umapi __gnu_pbds::gp_hash_table<ll, ll, xorshift>
#define umapp __gnu_pbds::gp_hash_table<P, ll, xorshift>
#define umapu __gnu_pbds::gp_hash_table<uint64_t, ll, xorshift>
#define umapip __gnu_pbds::gp_hash_table<ll, P, xorshift>

#endif
/*@formatter:on*/
struct xorshift {
  static uint64_t splitmix64(uint64_t x) {
    x += 0x9e3779b97f4a7c15;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
  }
  size_t operator()(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()(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);
  }
};
/*@formatter:off*/
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);
}
// 衝突対策
#define ws ws_

template <class A, class B, class C> struct T2 {
  A f;
  B s;
  C t;
  T2() { f = 0, s = 0, t = 0; }
  T2(A f, B s, C t) : f(f), s(s), t(t) {}
  bool operator<(const T2 &r) const {
    return f != r.f   ? f < r.f
           : s != r.s ? s < r.s
                      : t < r.t; /*return f != r.f ? f > r.f : s != r.s ?n s >
                                    r.s : t > r.t; 大きい順 */
  }
  bool operator>(const T2 &r) const {
    return f != r.f   ? f > r.f
           : s != r.s ? s > r.s
                      : t > r.t; /*return f != r.f ? f > r.f : s != r.s ? s >
                                    r.s : t > r.t; 小さい順 */
  }
  bool operator==(const T2 &r) const {
    return f == r.f && s == r.s && t == r.t;
  }
  bool operator!=(const T2 &r) const {
    return f != r.f || s != r.s || t != r.t;
  }
};
template <class A, class B, class C, class D> struct F2 {
  A a;
  B b;
  C c;
  D d;
  F2() { a = 0, b = 0, c = 0, d = 0; }
  F2(A a, B b, C c, D d) : a(a), b(b), c(c), d(d) {}
  bool operator<(const F2 &r) const {
    return a != r.a   ? a < r.a
           : b != r.b ? b < r.b
           : c != r.c ? c < r.c
                      : d < r.d; /*    return a != r.a ? a > r.a : b != r.b ? b
                                    > r.b : c != r.c ? c > r.c : d > r.d;*/
  }
  bool operator>(const F2 &r) const {
    return a != r.a   ? a > r.a
           : b != r.b ? b > r.b
           : c != r.c ? c > r.c
                      : d > r.d; /*        return a != r.a ? a < r.a : b != r.b
                                    ? b < r.b : c != r.c ? c < r.c : d < r.d;*/
  }
  bool operator==(const F2 &r) const {
    return a == r.a && b == r.b && c == r.c && d == r.d;
  }
  bool operator!=(const F2 &r) const {
    return a != r.a || b != r.b || c != r.c || d != r.d;
  }
  ll operator[](ll i) {
    assert(i < 4);
    return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;
  }
};
typedef T2<ll, ll, ll> T;
typedef F2<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); }

//@マクロ省略系 型,構造
#define double long double
using dou = double;
const double eps = 1e-9;
// 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) { 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; }};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 b, epsdou a){return a.v o
(dou)b;} #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(/)
#undef double
#define double epsdou
*/

#define ull unsigned long long
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;
using mvp = mvec<P>;
using mvt = mvec<T>;
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()

#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; }

// 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 vvi(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(ll, __VA_ARGS__)
#define vvb(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(bool, __VA_ARGS__)
#define vvs(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(string, __VA_ARGS__)
#define vvd(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(double, __VA_ARGS__)
#define vvc(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(char, __VA_ARGS__)
#define vvp(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(P, __VA_ARGS__)
#define vvt(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(T, __VA_ARGS__)
#define vv(type, ...)                                                          \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(type, __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
// 任意のマクロサポート用 使う度に初期化する
int index_, v1_, v2_, v3_;

template <class T> struct pq {
  priority_queue<T, vector<T>, greater<T>> q; /*小さい順*/
  T su = 0;
  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> struct pqg {
  priority_queue<T> q; /*大きい順*/
  T su = 0;
  void clear() {
    q = priority_queue<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(); }
};
#define pqi pq<ll>
#define pqgi pqg<ll>
// マクロ 繰り返し
// ↓@オーバーロード隔離

#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)
// 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 I = (int)a.size(), J = (int)a[0].size(), i = 0, j = 0; i < I;       \
       nex_repv2(i, j, I, J, 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 I = (int)a.size(), J = (int)a[0].size(), K = (int)a[0][0].size(),   \
           i = 0, j = 0, k = 0;                                                \
       i < I; nex_repv3(i, j, k, I, J, K, 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]))

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 form1(st)                                                              \
  for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form3(k, v, st)                                                        \
  for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form4(k, v, st, r)                                                     \
  for (auto &&form_it = st.begin(); form_it != st.end() && (*form_it).fi < r;  \
       ++form_it)
#define form5(k, v, st, l, r)                                                  \
  for (auto &&form_it = st.lower_bound(l);                                     \
       form_it != st.end() && (*form_it).fi < r; ++form_it)

#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)

#define fors1(st)                                                              \
  for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors2(v, st)                                                           \
  for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors3(v, st, r)                                                        \
  for (auto &&fors_it = st.begin(); fors_it != st.end() && (*fors_it) < r;     \
       ++fors_it)
#define fors4(v, st, l, r)                                                     \
  for (auto &&fors_it = st.lower_bound(l);                                     \
       fors_it != st.end() && (*fors_it) < r; ++fors_it)

#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)

template <class U> vector<U> to1d(vector<U> &a) { return a; }
template <class U> vector<U> 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 fora_init_2(a, A) ;
#define fora_init_3(fora_i, a, A) auto &&a = A[fora_i];
#define fora_init_4(a, b, A, B)                                                \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];
#define fora_init_5(fora_i, a, b, A, B)                                        \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];
#define fora_init_6(a, b, c, A, B, C)                                          \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];
#define fora_init_7(fora_i, a, b, c, A, B, C)                                  \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];
#define fora_init_8(a, b, c, d, A, B, C, D)                                    \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];                                                        \
  auto &&d = D[fora_i];
#define fora_init_9(fora_i, a, b, c, d, A, B, C, D)                            \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];                                                        \
  auto &&d = D[fora_i];
#define fora_init(...)                                                         \
  over9(__VA_ARGS__, fora_init_9, fora_init_8, fora_init_7, fora_init_6,       \
        fora_init_5, fora_init_4, fora_init_3, fora_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_init(k, v, ...)                                                   \
  auto &&k = (*forp_it).fi;                                                    \
  auto &&v = (*forp_it).se;
#define form_init(k, v, ...)                                                   \
  auto &&k = (*form_it).fi;                                                    \
  auto &&v = (*form_it).se;
#define forrm_init(k, v, ...)                                                  \
  auto &&k = (*forrm_it).fi;                                                   \
  auto &&v = (*forrm_it).se;
#define fors_init(v, ...) auto &&v = (*fors_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__);

#define fora_2(a, A) for (auto &&a : A)
#define fora_3(fora_i, a, A) rep(fora_i, sz(A))
#define fora_4(a, b, A, B) rep(fora_i, sz(A))
#define fora_5(fora_i, a, b, A, B) rep(fora_i, sz(A))
#define fora_6(a, b, c, A, B, C) rep(fora_i, sz(A))
#define fora_7(fora_i, a, b, c, A, B, C) rep(fora_i, sz(A))
#define fora_8(a, b, c, d, A, B, C, D) rep(fora_i, sz(A))
#define fora_9(fora_i, a, b, c, d, A, B, C, D) rep(fora_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)

// ↑@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__)
// 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)

#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(...)                                                              \
  over9(__VA_ARGS__, forr_9, forr_8, forr_7, forr_6, forr_5, forr_4, forr_3,   \
        forr_2)(__VA_ARGS__)

// #define forr(v, a) for(auto&& v : makeReversedRange(a))
// 参照を取らない
#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))

#define form(...)                                                              \
  over5(__VA_ARGS__, form5, form4, form3, form2, form1)(__VA_ARGS__)
#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 forp(k, v, st)                                                         \
  for (auto &&forp_it = st.begin(); forp_it != st.end(); ++forp_it)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010
const ll inf = (ll)1e9 + 100;
const ll linf = (ll)1e18 + 100;
const dou dinf = (dou)linf * linf;
const char infc = '{';
const string infs = "{";
const double PI = 3.1415926535897932384626433832795029L;

// 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 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()

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:on*/
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 << p.f << " " << p.s << " " << p.t;
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << p.a << " " << p.b << " " << p.c << " " << p.d;
  return os;
}
template <typename T> ostream &operator<<(ostream &os, 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, 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, map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}
template <class T> ostream &operator<<(ostream &os, set<T> s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, mset<T> s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, deque<T> a) {
  fora(v, a) os << v << " ";
  return os;
}
ostream &operator<<(ostream &os, vector<vector<char>> &vec) {
  rep(h, sz(vec)) {
    rep(w, sz(vec[0])) { os << vec[h][w]; }
    os << endl;
  }
  return os;
}
// template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>&
// a) {fora(v,a)os<<v<<endl;return os;}
/*@formatter:off*/
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...);
}
// template<typename W, typename H> void resize(vector<W> &vec, const H head) {
// vec.resize(head); } template<typename W, typename H, typename ... T> void
// resize(vector<W> &vec, const H &head, const T ... tail) {vec.resize(head);for
// (auto &v: vec)resize(v, tail...);}
template <typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool all_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (!all_of2(v[i], f))
      return false;
  }
  return true;
}
template <typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool any_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (any_of2(v[i], f))
      return true;
  }
  return false;
}
template <typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool none_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (none_of2(v[i], f))
      return false;
  }
  return true;
}
template <typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll find_if2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (find_if2(v[i], f))
      return i;
  }
  return sz(v);
}
template <typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll rfind_if2(vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (rfind_if2(v[i], f))
      return i;
  }
  return -1;
}
template <class T> bool contains(string &s, const T &v) {
  return s.find(v) != string::npos;
}
template <typename T> bool contains(vector<T> &v, const T &val) {
  return std::find(v.begin(), v.end(), val) != v.end();
}
template <typename T, typename F> bool contains_if2(vector<T> &v, F f) {
  return find_if(v.begin(), v.end(), f) != v.end();
}
template <typename T, typename F> ll count_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll count_if2(vector<T> &vec, F f) {
  ll ret = 0;
  fora(v, vec) ret += count_if2(v, f);
  return ret;
}
template <typename T, typename F> void for_each2(T &v, F f) { f(v); }
template <typename T, typename F> void for_each2(vector<T> &vec, F f) {
  fora(v, vec) for_each2(v, f);
}
template <typename W> ll count_od(vector<W> &a) {
  return count_if2(a, [](ll v) { return v & 1; });
}
template <typename W> ll count_ev(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(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(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> l_erase_if2(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> ll l_rfind_if(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(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(...) over3(__VA_ARGS__, 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)

#define for_each_s__2(a, right)                                                \
  do {                                                                         \
    fora(v, a) { v right; }                                                    \
  } while (0)
#define for_each_s__3(a, v, siki)                                              \
  do {                                                                         \
    fora(v, a) { siki; }                                                       \
  } while (0)
#define for_each_s(...)                                                        \
  over3(__VA_ARGS__, for_each_s__3, for_each_s__2)(__VA_ARGS__)

#define for_each__2(a, right) for_each2(a, lamr(right))
#define for_each__3(a, v, siki) for_each2(a, [&](auto v) { return siki; })
#define for_each(...) over3(__VA_ARGS__, for_each__3, for_each__2)(__VA_ARGS__)
#define for_each_f(a, f) for_each2(a, f);

#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)

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());
}
void replace(str &a, char key, char v) { replace(all(a), key, v); }
// 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;
}
// template<class T> void replace(T &a) { replace(a, '#'); }
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(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 includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(all(c));
  sort(all(d));
  return includes(all(c), all(d));
}
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));
}
void iota(vector<ll> &ve, ll s, ll n) {
  ve.resize(n);
  iota(all(ve), s);
}
vi iota(ll s, ll len) {
  vi ve(len);
  iota(all(ve), s);
  return ve;
}
template <class A, class B> auto vtop(vector<A> &a, vector<B> &b) {
  assert(sz(a) == sz(b)); /*stringを0で初期化できない  */
  vector<pair<A, B>> res;
  rep(i, sz(a)) res.eb(a[i], b[i]);
  return res;
}
template <class A, class B>
void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {
  a.resize(sz(p)), b.resize(sz(p));
  rep(i, sz(p)) a[i] = p[i].fi, b[i] = p[i].se;
}
template <class A, class B, class C>
auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {
  assert(sz(a) == sz(b) && sz(b) == sz(c));
  vector<T2<A, B, C>> res;
  rep(i, sz(a)) res.eb(a[i], b[i], c[i]);
  return res;
}
template <class A, class B, class C, class D>
auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));
  vector<F2<A, B, C, D>> res;
  rep(i, sz(a)) res.eb(a[i], b[i], c[i], d[i]);
  return res;
}
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {
  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
};
template <class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {
  typedef pair<A, B> U;
  if (type == fisi)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });
  else if (type == fisd)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });
  else if (type == fdsi)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });
  else if (type == fdsd)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });
  else if (type == sifi)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });
  else if (type == sifd)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });
  else if (type == sdfi)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });
  else if (type == sdfd)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
};
template <class U> void sort(vector<U> &a, pcomparator type) {
  if (type == fisi)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });
  else if (type == fisd)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });
  else if (type == fdsi)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });
  else if (type == fdsd)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });
  else if (type == sifi)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });
  else if (type == sifd)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });
  else if (type == sdfi)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });
  else if (type == sdfd)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });
};
template <class A, class B, class C, class D>
void sort(vector<F2<A, B, C, D>> &a, pcomparator type) {
  typedef F2<A, B, C, D> U;
  if (type == fisi)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; });
  else if (type == fisd)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; });
  else if (type == fdsi)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; });
  else if (type == fdsd)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; });
  else if (type == sifi)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; });
  else if (type == sifd)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; });
  else if (type == sdfi)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; });
  else if (type == sdfd)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });
};
template <class U> void sort(vector<U> &a, tcomparator type) {
  if (type == 0)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t;
    });
  else if (type == 1)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t;
    });
  else if (type == 2)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t;
    });
  else if (type == 3)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t;
    });
  else if (type == 4)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t;
    });
  else if (type == 5)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t;
    });
  else if (type == 6)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t;
    });
  else if (type == 7)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t;
    });
  else if (type == 8)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s;
    });
  else if (type == 9)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s;
    });
  else if (type == 10)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s;
    });
  else if (type == 11)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s;
    });
  else if (type == 12)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s;
    });
  else if (type == 13)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s;
    });
  else if (type == 14)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s;
    });
  else if (type == 15)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s;
    });
  else if (type == 16)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t;
    });
  else if (type == 17)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t;
    });
  else if (type == 18)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t;
    });
  else if (type == 19)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t;
    });
  else if (type == 20)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t;
    });
  else if (type == 21)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t;
    });
  else if (type == 22)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t;
    });
  else if (type == 23)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t;
    });
  else if (type == 24)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f;
    });
  else if (type == 25)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f;
    });
  else if (type == 26)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f;
    });
  else if (type == 27)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f;
    });
  else if (type == 28)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f;
    });
  else if (type == 29)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f;
    });
  else if (type == 30)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f;
    });
  else if (type == 31)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f;
    });
  else if (type == 32)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s;
    });
  else if (type == 33)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s;
    });
  else if (type == 34)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s;
    });
  else if (type == 35)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s;
    });
  else if (type == 36)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s;
    });
  else if (type == 37)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s;
    });
  else if (type == 38)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s;
    });
  else if (type == 39)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s;
    });
  else if (type == 40)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f;
    });
  else if (type == 41)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f;
    });
  else if (type == 42)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f;
    });
  else if (type == 43)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f;
    });
  else if (type == 44)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f;
    });
  else if (type == 45)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f;
    });
  else if (type == 46)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f;
    });
  else if (type == 47)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f;
    });
}
template <class A, class B, class C, class D>
void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {
  typedef F2<A, B, C, D> U;
  if (type == 0)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c;
    });
  else if (type == 1)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c;
    });
  else if (type == 2)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c;
    });
  else if (type == 3)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c;
    });
  else if (type == 4)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c;
    });
  else if (type == 5)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c;
    });
  else if (type == 6)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c;
    });
  else if (type == 7)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c;
    });
  else if (type == 8)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b;
    });
  else if (type == 9)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b;
    });
  else if (type == 10)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b;
    });
  else if (type == 11)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b;
    });
  else if (type == 12)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b;
    });
  else if (type == 13)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b;
    });
  else if (type == 14)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b;
    });
  else if (type == 15)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b;
    });
  else if (type == 16)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c;
    });
  else if (type == 17)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c;
    });
  else if (type == 18)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c;
    });
  else if (type == 19)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c;
    });
  else if (type == 20)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c;
    });
  else if (type == 21)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c;
    });
  else if (type == 22)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c;
    });
  else if (type == 23)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c;
    });
  else if (type == 24)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a;
    });
  else if (type == 25)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a;
    });
  else if (type == 26)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a;
    });
  else if (type == 27)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a;
    });
  else if (type == 28)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a;
    });
  else if (type == 29)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a;
    });
  else if (type == 30)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a;
    });
  else if (type == 31)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a;
    });
  else if (type == 32)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b;
    });
  else if (type == 33)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b;
    });
  else if (type == 34)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b;
    });
  else if (type == 35)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b;
    });
  else if (type == 36)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b;
    });
  else if (type == 37)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b;
    });
  else if (type == 38)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b;
    });
  else if (type == 39)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b;
    });
  else if (type == 40)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a;
    });
  else if (type == 41)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a;
    });
  else if (type == 42)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a;
    });
  else if (type == 43)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a;
    });
  else if (type == 44)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a;
    });
  else if (type == 45)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a;
    });
  else if (type == 46)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a;
    });
  else if (type == 47)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a;
    });
}

void sort(string &a) { sort(all(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> void sort(vector<T> &a) { sort(all(a)); }
// P l, P rで f(P) の形で渡す
template <class U, class F> void sort(vector<U> &a, F f) {
  sort(all(a), [&](U l, U r) { return f(l) < f(r); });
};
template <class T> void rsort(vector<T> &a) { sort(all(a), greater<T>()); };
template <class U, class F> void rsort(vector<U> &a, F f) {
  sort(all(a), [&](U l, U r) { return f(l) > f(r); });
};

// F = T<T>
// 例えばreturn p.fi + p.se;
template <class A, class B> void sortp(vector<A> &a, vector<B> &b) {
  auto c = vtop(a, b);
  sort(c);
  rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;
}
template <class A, class B, class F>
void sortp(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  sort(c, f);
  rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;
}
template <class A, class B> void rsortp(vector<A> &a, vector<B> &b) {
  auto c = vtop(a, b);
  rsort(c);
  rep(i, sz(a)) a[i] = c[i].first, b[i] = c[i].second;
}
template <class A, class B, class F>
void rsortp(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  rsort(c, f);
  rep(i, sz(a)) a[i] = c[i].first, b[i] = c[i].second;
}
template <class A, class B, class C>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c) {
  auto d = vtot(a, b, c);
  sort(d);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class F>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  sort(d, f);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c) {
  auto d = vtot(a, b, c);
  rsort(d);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class F>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  rsort(d, f);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class D>
void sortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  auto e = vtof(a, b, c, d);
  sort(e);
  rep(i, sz(a)) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
template <class A, class B, class C, class D>
void rsortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  auto e = vtof(a, b, c, d);
  rsort(e);
  rep(i, sz(a)) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
// sortindex 元のvectorはソートしない
template <class T> vi sorti(vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(a));
  sortp(b, ind);
  return ind;
} /*indexの分で型が変わるためpcomparatorが必要*/
template <class T> vi sorti(vector<T> &a, pcomparator f) {
  auto b = a;
  vi ind = iota(0, sz(a));
  sortp(b, ind, f);
  return ind;
}
template <class T, class F> vi sorti(vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(a[x]) < f(a[y]); });
  return ind;
}
template <class T> vi rsorti(vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(a));
  rsortp(b, ind);
  return ind;
}
template <class T, class F> vi rsorti(vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(a[x]) > f(a[y]); });
  return ind;
}
template <class A, class B, class F>
vi sortpi(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(c[x]) < f(c[y]); });
  return ind;
}
template <class A, class B>
vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {
  vi ind = iota(0, sz(a));
  auto c = a;
  auto d = b;
  sortt(c, d, ind, f);
  return ind;
}
template <class A, class B> vi sortpi(vector<A> &a, vector<B> &b) {
  return sortpi(a, b, fisi);
};
template <class A, class B, class F>
vi rsortpi(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(c[x]) > f(c[y]); });
  return ind;
}
template <class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) {
  return sortpi(a, b, fdsd);
};
template <class A, class B, class C, class F>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(d[x]) < f(d[y]); });
  return ind;
}
template <class A, class B, class C>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {
  vi ind = iota(0, sz(a));
  auto d = vtof(a, b, c, ind);
  sort(d, f);
  rep(i, sz(a)) ind[i] = d[i].d;
  return ind;
}
template <class A, class B, class C>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) {
    if (a[x] == a[y]) {
      if (b[x] == b[y])
        return c[x] < c[y];
      else
        return b[x] < b[y];
    } else {
      return a[x] < a[y];
    }
  });
  return ind;
}
template <class A, class B, class C, class F>
vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(d[x]) > f(d[y]); });
  return ind;
}
template <class A, class B, class C>
vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) {
    if (a[x] == a[y]) {
      if (b[x] == b[y])
        return c[x] > c[y];
      else
        return b[x] > b[y];
    } else {
      return a[x] > a[y];
    }
  });
  return ind;
}
template <class T> void sort2(vector<vector<T>> &a) {
  for (ll i = 0, n = a.size(); i < n; ++i)
    sort(a[i]);
}
template <class T> void rsort2(vector<vector<T>> &a) {
  for (ll i = 0, n = a.size(); i < n; ++i)
    rsort(a[i]);
}

template <class... T, class U> auto sorted(U head, T... a) {
  sort(head, a...);
  return head;
}

template <typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) {
  rep(i, N) a[i] = v;
}
template <typename A, size_t N, size_t O, typename T>
void fill(A (&a)[N][O], const T &v) {
  rep(i, N) rep(j, O) a[i][j] = v;
}
template <typename A, size_t N, size_t O, size_t P, typename T>
void fill(A (&a)[N][O][P], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) a[i][j][k] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, typename T>
void fill(A (&a)[N][O][P][Q], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) a[i][j][k][l] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          typename T>
void fill(A (&a)[N][O][P][Q][R], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) a[i][j][k][l][m] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S, typename T>
void fill(A (&a)[N][O][P][Q][R][S], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S)
      a[i][j][k][l][m][n] = v;
}
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, ll len, const T v) {
  rep(i, len) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<vector<W>> &xx, int sh, int th, int sw, int tw, T v) {
  rep(h, sh, th) rep(w, sw, tw) xx[h][w] = v;
}
template <class T, class U> void fill(vector<T> &a, vi &ind, U val) {
  fora(v, ind) a[v] = val;
}
template <class W, class T> void fill(mvec<W> &xx, const T v) {
  fora(x, xx) fill(x, v);
}

template <typename A, size_t N> A sum(A (&a)[N]) {
  A res = 0;
  rep(i, N) res += a[i];
  return res;
}
template <typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {
  A res = 0;
  rep(i, N) rep(j, O) res += a[i][j];
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) res += a[i][j][k];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A sum(A (&a)[N][O][P][Q]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) res += a[i][j][k][l];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A sum(A (&a)[N][O][P][Q][R]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) res += a[i][j][k][l][m];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A sum(A (&a)[N][O][P][Q][R][S]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S) res +=
      a[i][j][k][l][m][n];
  return res;
}
//@汎用便利関数 入力
ll in() {
  ll 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...);
}

#define din1(a)                                                                \
  ll a;                                                                        \
  cin >> a
#define din2(a, b)                                                             \
  ll a, b;                                                                     \
  cin >> a >> b
#define din3(a, b, c)                                                          \
  ll a, b, c;                                                                  \
  cin >> a >> b >> c
#define din4(a, b, c, d)                                                       \
  ll a, b, c, d;                                                               \
  cin >> a >> b >> c >> d
#define din5(a, b, c, d, e)                                                    \
  ll a, b, c, d, e;                                                            \
  cin >> a >> b >> c >> d >> e
#define din6(a, b, c, d, e, f)                                                 \
  ll a, b, c, d, e, f;                                                         \
  cin >> a >> b >> c >> d >> e >> f
#define din(...)                                                               \
  over6(__VA_ARGS__, din6, din5, din4, din3, din2, din1)(__VA_ARGS__)

#define dins1(a)                                                               \
  str a;                                                                       \
  cin >> a
#define dins2(a, b)                                                            \
  str a, b;                                                                    \
  cin >> a >> b
#define dins3(a, b, c)                                                         \
  str a, b, c;                                                                 \
  cin >> a >> b >> c
#define dins4(a, b, c, d)                                                      \
  str a, b, c, d;                                                              \
  cin >> a >> b >> c >> d
#define dins5(a, b, c, d, e)                                                   \
  str a, b, c, d, e;                                                           \
  cin >> a >> b >> c >> d >> e
#define dins6(a, b, c, d, e, f)                                                \
  str a, b, c, d, e, f;                                                        \
  cin >> a >> b >> c >> d >> e >> f
#define dins(...)                                                              \
  over6(__VA_ARGS__, dins6, dins5, dins4, dins3, dins2, dins1)(__VA_ARGS__)

#define din1d(a)                                                               \
  din1(a);                                                                     \
  a--
#define din2d(a, b)                                                            \
  din2(a, b);                                                                  \
  a--, b--
#define din3d(a, b, c)                                                         \
  din3(a, b, c);                                                               \
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \
  din4(a, b, c, d);                                                            \
  a--, b--, c--, d--
#define dind(...) over4(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

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; }

#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; }
template <class T> string out_m2(vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), 12ll);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) {
    ss << a[i];
    if (typeid(a[i]) == typeid(P)) {
      ss << endl;
    } else {
      ss << " ";
    }
  }
  return ss.str();
}
template <class T>
string out_m2(vector<vector<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  H = min({H, sz(a), 12ll});
  W = min({W, sz(a[0]), 12ll});
  stringstream ss;
  ss << endl;
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  rep(w, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 4 + 3) ss << "_";
  ss << "" << endl;
  rep(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    rep(w, min(sz(a[h]), 12ll)) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  return ss.str();
}
template <class T>
string out_m2(vector<vector<vector<T>>> &a, ll H = inf, ll W = inf,
              ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 12;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << out_m2(a[i], W, U, i);
  }
  return ss.str();
}
template <class T, size_t N> string out_m2(T (&a)[N]) {
  vector<T> b;
  resize(b, N);
  rep(i, N) { b[i] = a[i]; }
  return out_m2(b);
}
template <class T, size_t N, size_t M> string out_m2(T (&a)[N][M]) {
  vector<vector<T>> b;
  resize(b, N, M);
  rep(i, N) {
    rep(j, M) { b[i][j] = a[i][j]; }
  }
  return out_m2(b);
}
template <class T, size_t N, size_t M, size_t O>
string out_m2(T (&a)[N][M][O]) {
  vector<vector<vector<T>>> b;
  resize(b, N, M, O);
  rep(i, N) {
    rep(j, M) {
      rep(k, O) { b[i][j][k] = a[i][j][k]; }
    }
  }
  return out_m2(b);
}
string out_m2(int a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
/*@formatter:on*/
template <class T>
string out_m2(mvec<mvec<T>> &a, ll H = inf, ll W = inf, int key = inf) {
  H = min({H, sz(a), 6ll});
  W = min({W, sz(a[0]), 6ll});
  stringstream ss;
  ss << endl;
  //    if (key == inf)ss << " *|"; else ss << " " << key << "|";
  if (key == inf)
    ss << " *|";
  else {
    ss << std::right << std::setw(2) << key;
    ss << "|";
  }
  rep(w, -W, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 8 + 3) ss << "_";
  ss << "" << endl;
  rep(h, -H, H) {
    ss << std::right << std::setw(2) << h << "|";
    int NW = min(sz(a[h]), 6ll);
    rep(w, -NW, NW) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  return ss.str();
} /*@formatter:on*/
template <class T>
string out_m2(mvec<mvec<mvec<T>>> &a, ll H = inf, ll W = inf, ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 6;
  H = min(H, sz(a));
  rep(i, -H, H) {
    ss << endl;
    ss << out_m2(a[i], W, U, i);
  }
  return ss.str();
}

/*@formatter:off*/
template <class T> string out_m2(T &a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
/*@formatter:on*/
template <class T> string out_m(vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), 12ll);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) { ss << a[i] << " "; }
  ss << "" << endl;
  return ss.str();
}
/*@formatter:off*/
template <class T>
string out_m(vector<vector<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  H = min({H, sz(a), 12ll});
  W = min({W, sz(a[0]), 12ll});
  stringstream ss;
  ss << endl;
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  rep(w, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 4 + 3) ss << "_";
  ss << "" << endl;
  rep(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    rep(w, min(sz(a[h]), 12ll)) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  ss << endl;
  return ss.str();
}
template <class T>
string out_m(vector<vector<vector<T>>> &a, ll H = inf, ll W = inf, ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 5;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << out_m(a[i], W, U, i);
  }
  ss << endl;
  return ss.str();
}
string out_m(int a) {
  stringstream ss;
  ss << a << endl;
  return ss.str();
}
template <class T> string out_m(T &a) {
  stringstream ss;
  ss << a << endl;
  return ss.str();
}
template <class T> void outv(vector<T> &a, ll W = inf) {
  cout << out_m(a, W) << endl;
}
template <class T>
void outv(vector<vector<T>> &a, ll H = linf, ll W = linf, int key = -1) {
  cout << out_m(a, H, W, key) << endl;
}
template <class T>
void outv(vector<vector<vector<T>>> &a, ll H = linf, ll W = linf, ll U = linf) {
  cout << out_m(a, H, W, U) << endl;
}
#else
template <class T> void outv(vector<T> &a, ll W = inf) {
  rep(i, min(W, sz(a))) { cout << a[i] << " "; }
  cout << "" << endl;
}
template <class T>
void outv(vector<vector<T>> &a, ll H = linf, ll W = linf, int key = -1) {
  rep(i, min(H, sz(a))) { outv(a[i], W); }
}
template <class T>
void outv(vector<vector<vector<T>>> &a, ll H = linf, ll W = linf, ll U = linf) {
  ;
}
#define err(...) ;
#endif
template <class T> void outl(vector<T> &a, int n = inf) {
  rep(i, min(n, sz(a))) cout << a[i] << endl;
}
// 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; }
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(n);                                                                     \
  rep(dnai, n) cin >> a[dnai];
#define dnad(a, n)                                                             \
  vi a(n);                                                                     \
  rep(dnai, n) cin >> a[dnai], a[dnai]--;
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(n), b(n);                                                               \
  rep(dna2i, n) cin >> a[dna2i] >> b[dna2i];
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 dna2d(a, b, n)                                                         \
  vi a(n), b(n);                                                               \
  rep(dna2di, n) {                                                             \
    cin >> a[dna2di] >> b[dna2di];                                             \
    a[dna2di]--, b[dna2di]--;                                                  \
  }
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]--;
}
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(n), b(n), c(n);                                                         \
  rep(dna3i, n) cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
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(n), b(n), c(n);                                                         \
  rep(dna3di, n) {                                                             \
    cin >> a[dna3di] >> b[dna3di] >> c[dna3di];                                \
    a[dna3di]--, b[dna3di]--, c[dna3di]--;                                     \
  }
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(n), b(n), c(n), d(n);                                                   \
  rep(dna4i, n) cin >> a[dna4i] >> b[dna4i] >> c[dna4i] >> d[dna4i];
#define dna4d(a, b, c, d, n)                                                   \
  vi a(n), b(n), c(n), d(n);                                                   \
  rep(dna4i, n) cin >> a[dna4i] >> b[dna4i] >> c[dna4i] >> d[dna4i],           \
      --a[dna4i], --b[dna4i], --c[dna4i], --d[dna4i];
#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];
// デバッグ
#define sp << " " <<

#define deb1(x) debugName(x) << " = " << out_m2(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__)

#ifdef _DEBUG
#define deb(...)                                                               \
  cerr << over10(__VA_ARGS__, deb10, deb9, deb8, deb7, deb6, deb5, deb_4,      \
                 deb_3, deb_2, deb1)(__VA_ARGS__)                              \
       << endl

#define base_keta 8
void print_n_base(int x, int base) { cerr << bitset<base_keta>(x) << 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)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 2);
#define deb3(x)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 3);
#define deb4(x)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 4);

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

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

//@formatter:off
// よく使うクラス、構造体
// graphでredefineしているため、書き換えたら向こうも書き換える
struct unionfind {
  vector<ll> par;
  vector<ll> siz;
  vector<ll> es;
  ll n, trees; // 連結グループの数(親の種類)
  unionfind(ll n) : n(n), trees(n) {
    par.resize(n);
    siz.resize(n);
    es.resize(n);
    for (ll i = 0; i < n; i++) {
      par[i] = i;
      siz[i] = 1;
    }
  }
  ll root(ll x) {
    if (par[x] == x) {
      return x;
    } else {
      return par[x] = root(par[x]);
    }
  }
  ll operator()(ll x) { return root(x); }
  bool unite(ll x, ll y) {
    x = root(x);
    y = root(y);
    es[x]++;
    if (x == y)
      return false;
    if (siz[x] > siz[y])
      swap(x, y);
    trees--;
    par[x] = y;
    siz[y] += siz[x];
    es[y] += es[x];
    return true;
  }
  bool same(ll x, ll y) { return root(x) == root(y); }
  ll size(ll x) { return siz[root(x)]; }
  ll esize(ll x) { return es[root(x)]; }
  vi sizes() {
    vi cou(n);
    vi ret;
    ret.reserve(n);
    rep(i, n) { cou[root(i)]++; }
    rep(i, n) {
      if (cou[i])
        ret.push_back(cou[i]);
    }
    return ret;
  }
  // つながりを無向グラフと見なし、xが閉路に含まれるか判定
  bool close(ll x) { return esize(x) >= size(x); }
  vector<vi> sets() {
    vi ind(n, -1);
    ll i = 0;
    vvi(res, trees);
    rep(j, n) {
      ll r = root(j);
      if (ind[r] == -1)
        ind[r] = i++;
      res[ind[r]].push_back(j);
    }
    rep(i, trees) {
      ll r = root(res[i][0]);
      if (res[i][0] == r)
        continue;
      rep(j, 1, sz(res[i])) {
        if (res[i][j] == r) {
          swap(res[i][0], res[i][j]);
          break;
        }
      }
    }
    return res;
  }
}; //@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 < 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';
  }
}
bint max(bint a, signed b) { return max(a, (bint)b); }
bint max(signed a, bint b) { return max((bint)a, b); }
bint max(bint a, ll b) { return max(a, (bint)b); }
bint max(ll a, bint b) { return max((bint)a, b); }
bint min(bint a, signed b) { return min(a, (bint)b); }
bint min(signed a, bint b) { return min((bint)a, b); }
bint min(bint a, ll b) { return min(a, (bint)b); }
bint min(ll a, bint b) { return min((bint)a, b); }
// エラー
void ole() {
#ifdef _DEBUG
  debugline("ole");
  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() {
  assert(0 == 1);
  exit(0);
}
void tle() {
  while (inf)
    cout << inf << endl;
}

// 便利関数

// 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;
}
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

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 T> vector<pair<T, int>> run_length(vector<T> &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;
}
vector<pair<char, ll>> run_length(string &a) {
  vector<pair<char, ll>> 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:on*/
template <class T, class F> T mgr(T ok, T ng, F f) {
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
template <class T, class F> T mgr(signed ok_, T ng, F f) {
  T ok = ok_;
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
template <class T, class F> T mgr(T ok, signed ng_, F f) {
  T ng = ng_;
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
/*@formatter:off*/
template <class F> dou mgrd(dou ok, dou ng, F f, int kai = 100) {
  bool han = true;
  if (ok < ng)
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
// 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-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 -1;
}
template <typename W, typename T> ll find(vector<W> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return -1;
}
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>
T 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 mt(i, j, k);
  return mt(-1, -1, -1);
}
// stringも書く
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 -1;
}
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 -1;
}
int find(string &s, const char key) {
  rep(i, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return -1;
}
int find(string &s, int l, const char key) {
  rep(i, l, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return -1;
}
template <typename W, typename T> ll count2(W &a, const T k) { return a == k; }
template <typename W, typename T> ll count2(vector<W> &a, const T k) {
  ll ret = 0;
  fora(v, a) ret += count2(v, k);
  return ret;
}
template <typename W, typename T> ll count(vector<W> &a, const T k) {
  ll ret = 0;
  fora(v, a) ret += count2(v, k);
  return ret;
}
vi count(vi &a) {
  int ma = 0;
  fora(v, a) {
    if (ma < v)
      ma = v;
  }
  vi res(ma + 1);
  fora(v, a) { res[v]++; }
  return res;
}
ll count(str &a, 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;
}
//'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;
}
vi count(str &a, int r = inf) { return count(a, 0, min(r, sz(a))); }
#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> void rev(vector<T> &a) { reverse(all(a)); }
template <class U> void 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];
  a = b;
}
void rev(string &a) { reverse(all(a)); }
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) {
  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;
        }
      }
    }
  }
}
ll getr(ll a, ll keta) { return (a / (ll)pow(10, keta)) % 10; }
// 上から何桁目か
ll getl(ll a, ll ket) {
  int sketa = keta(a);
  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
//[v] := iとなるようなvectorを返す
// 存在しない物は-1
template <class T> auto keys(T &a) {
  vector<decltype((a.begin())->fi)> res;
  for (auto &&k : a)
    res.push_back(k.fi);
  return res;
}
template <class T> auto values(T &a) {
  vector<decltype((a.begin())->se)> res;
  for (auto &&k : a)
    res.push_back(k.se);
  return res;
}
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;
}
template <class T> constexpr T min(T a, signed b) { return a < b ? a : b; }
template <class T> constexpr T max(T a, signed b) { return a < b ? b : a; }
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;
}
template <class T> T min(vector<T> &a) { return *min_element(all(a)); }
template <class T> T mini(vector<T> &a) {
  return min_element(all(a)) - a.begin();
}
template <class T> T min(vector<T> &a, ll n) {
  return *min_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> T min(vector<T> &a, ll s, ll n) {
  return *min_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <class T> T max(vector<T> &a) { return *max_element(all(a)); }
template <class T, class U> T max(vector<T> &a, vector<U> &b) {
  return max(*max_element(all(a)), *max_element(all(b)));
}
template <class T> T maxi(vector<T> &a) {
  return max_element(all(a)) - a.begin();
}
template <class T> T max(vector<T> &a, ll n) {
  return *max_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> T max(vector<T> &a, ll s, ll n) {
  return *max_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <typename A, size_t N> A max(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = max(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> A max(A (&a)[N][O]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A max(A (&a)[N][O][P]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A max(A (&a)[N][O][P][Q], const T &v) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A max(A (&a)[N][O][P][Q][R]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A max(A (&a)[N][O][P][Q][R][S]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> A min(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = min(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> A min(A (&a)[N][O]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A min(A (&a)[N][O][P]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A min(A (&a)[N][O][P][Q], const T &v) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A min(A (&a)[N][O][P][Q][R]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A min(A (&a)[N][O][P][Q][R][S]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <class T> T sum(vector<T> &v, ll s, ll t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += v[i];
  return ret;
}
template <class T> T sum(vector<T> &v, ll t = inf) { return sum(v, 0, t); }
template <class T> T sum(vector<vector<T>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<T>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T> T sum(vector<vector<vector<T>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<vector<T>>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T> T sum(vector<vector<vector<vector<T>>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<vector<vector<T>>>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T>
T sum(vector<vector<vector<vector<vector<T>>>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T>
T sum(vector<vector<vector<vector<vector<T>>>>> &v, int t = inf) {
  return sum(v, 0, t);
}

template <class T> auto sum(priority_queue<T, vector<T>, greater<T>> &r) {
  auto q = r;
  T ret = 0;
  while (sz(q)) {
    ret += q.top();
    q.pop();
  }
  return ret;
}
template <class T> auto sum(priority_queue<T> &r) {
  auto q = r;
  T ret = 0;
  while (sz(q)) {
    ret += q.top();
    q.pop();
  }
  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;}
vi v_i(vi &a) {
  int n = max(a) + 1;
  vi ret(n, -1);
  rep(i, sz(a)) { 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;
}
// imoは0-indexed
// ruiは1-indexed
template <class T> vector<T> imo(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
// kと同じものの数
template <class T, class U> vi imo(vector<T> &a, U k) {
  vector<T> ret = a;
  rep(i, sz(ret)) ret[i] = a[i] == k;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
template <class T> vector<T> imox(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(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(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(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(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 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) { 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]; }
  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> 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<T> ri;
  bool build = false;

public:
  int n;
  ruimax(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    int ma = MIN(T);
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = -linf;
    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 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;
    }
  }
  T geti(int r = inf) { return geti(0, min(r, n)); };
  T getv(int r = inf) { return getv(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<T> ri;
  bool build = false;
  int n;

public:
  ruimin(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    int mi = MAX(T);
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = linf;
    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 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;
    }
  }
  T geti(int r = inf) { return geti(0, min(r, n)); };
  T getv(int r = inf) { return getv(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};

template <class T> ostream &operator<<(ostream &os, ruiC<T> a) {
  fora(v, a.rui) os << v << " ";
  return os;
}
template <class T> vector<T> ruiv(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() { return ruiC<T>(); }
template <class T> ruiC<T> ruic(vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
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;
}
vector<ll> ruiv(string &a) {
  if (sz(a) == 0)
    return vi(1);
  ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;
  vector<ll> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i] - dec;
  return ret;
}
ruiC<ll> ruic(string &a) {
  vector<ll> ret = ruiv(a);
  return ruiC<ll>(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;
  ruiC2(vector<vector<T>> &ru) : rui(sz(ru)), H(sz(ru)) {
    for (int h = 0; h < H; 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) {
    assert(h < H);
    return rui[h];
  }
  //    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();}
};

// a~zを0~25として
//  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
ruiC2<ll> ruicou(str &a) {
  str s = a;
  replace(s);
  vector<ruiC<ll>> res(26);
  vvi(cou, 26, sz(s));
  rep(i, sz(s)) { cou[s[i]][i] = 1; }
  return ruiC2<ll>(cou);
}
ruiC2<ll> ruicou(vi &a) {
  int H = max(a) + 1;
  vector<ruiC<ll>> res(H);
  vvi(cou, H, sz(a));
  rep(i, 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);
}
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;
}

#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);
}
/*@formatter:off*/
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> 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 ins(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 ins(0, i, sz(a)) && ins(0, j, sz(a));
}
#define inside ins
ll u(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u(vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) v = u(v);
  return ret;
}

// 添え字を返す
template <class F> ll 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 resIndex;
}
template <class F> ll 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 resIndex;
}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template <class F> dou 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 left;
}
template <class F> dou 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 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> ll 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 mini;
}
template <class F> ll 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 maxi;
}
template <class F>
dou 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 mini;
}
template <class F>
dou 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 maxi;
}
template <class T> T min(vector<vector<T>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, *min_element(all(a[i])));
  return res;
}
template <class T> T max(vector<vector<T>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, *max_element(all(a[i])));
  return res;
}
template <class T> T min(vector<vector<vector<T>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<T>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<T>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<T>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<vector<T>>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<vector<T>>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
// pow周りの仕様
// powiを使うと整数型
// powbを使うとbint型
// powを使うと powlに変換され long doubleが返る
#ifdef _DEBUG
auto pow(ll a, ll k) {
  static bool was = 1;
  if (was) {
    message += "if integer use *powi* it's very fast\n";
  }
  was = 0;
  return powl(a, k);
}
// 上のメッセージを出すための関数
auto pow(signed a, ll k) { return pow((ll)a, k); }
auto pow(signed a, signed k) { return pow((ll)a, (ll)k); }
#else
#define pow powl

#endif
// 整数型のpow
int powi(int a, int k) {
  if (a == 2)
    return 1ll << k;
  int res = 1;
  int x = a;
  while (k) {
    if (k & 1)
      res *= x;
    x *= x;
    k >>= 1;
  }
  return res;
}
// define pow powlより上に動かすとバグる
bint pow(bint a, ll k) {
  bint res = 1;
  bint x = a;
  while (k) {
    if (k & 1)
      res *= x;
    x *= x;
    k >>= 1;
  }
  return res;
}
bint pow(bint a, signed k) { return pow(a, (ll)k); }
bint powb(int a, int b) { return pow((bint)a, b); }
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;
}
ll bget(ll m, ll keta, ll sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
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));
}
ll bit(ll n, ll sinsuu) { return (ll)pow(sinsuu, n); }
ll mask(ll n) { return (1ll << n) - 1; }
// aをbitに置きなおす
//{0, 2} -> 101
ll bit(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 bitb(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) { return n & -n; }
ll lbiti(ll n) { return log2(n & -n); }
// 最上位ビット
ll hbit(ll 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) { 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;
}
// masの立ってるindexを見る
#define forbit(i, mas)                                                         \
  for (int forbitj = lbit(mas), forbitm = mas, i = log2(forbitj); forbitm;     \
       forbitm = forbitj ? forbitm ^ forbitj : 0, forbitj = lbit(forbitm),     \
           i = log2(forbitj))

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;
  }
  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 unique(v) v.erase(unique(v.begin(), v.end()), v.end());
//[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;
}
vi compress(vi &a, umap<ll, ll> &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();
  }
}
// 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 lowerBound(ruiC<T> &a, U v, W banpei) {
  return lowerBound(a.rui, v, banpei);
}
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 upperBound(ruiC<T> &a, U v, W banpei) {
  return upperBound(a.rui, v, banpei);
}
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 rlowerBound(ruiC<T> &a, U v, W banpei) {
  return rlowerBound(a.rui, v, banpei);
}
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);
  }
}
template <class T, class U, class W> T rupperBound(ruiC<T> &a, U v, W banpei) {
  return rupperBound(a.rui, v, banpei);
}
#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; };
ll fact(ll v) { return v <= 1 ? 1 : v * fact(v - 1); }
dou factd(int v) {
  static vd fact(2, 1);
  if (sz(fact) <= v) {
    rep(i, sz(fact), v + 1) { fact.push_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];
}
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];
}

ll gcd(ll a, ll b) {
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
ll gcd(vi b) {
  ll res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
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;
  }
}

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); }

// 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;
}

constexpr int bsetlen = k5 * 2;
// 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> 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, class U> void operator+=(pair<T, U> &a, pair<T, U> &b) {
  a.fi += b.fi;
  a.se += b.se;
}
template <class T, class U> pair<T, U> operator+(pair<T, U> &a, pair<T, U> &b) {
  return pair<T, U>(a.fi + b.fi, a.se + b.se);
}

template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  basic_string<CharT, Traits, Alloc> str(lhs);
  str.append(to_string(rv));
  return str;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  lhs += to_string(rv);
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  basic_string<CharT, Traits, Alloc> str(lhs);
  str.append(to_string(rv));
  return str;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  lhs += to_string(rv);
}
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, U v) { a.push(v); }
template <class T, class U> void operator+=(deque<T> &a, 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, U v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, U v) {
  a.push(v);
  return a;
}
template <class T> set<T> &operator+=(set<T> &a, vector<T> v) {
  fora(d, v) a.insert(d);
  return a;
}
template <class T, class U> auto operator+=(set<T> &a, U v) {
  return a.insert(v);
}
template <class T, class U> auto operator-=(set<T> &a, U v) {
  return a.erase(v);
}
template <class T, class U> auto operator+=(mset<T> &a, U v) {
  return a.insert(v);
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, U v) {
  a.push_back(v);
  return a;
}
template <class T, class U> vector<T> operator+(vector<T> &a, U v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> vector<T> operator+(U v, vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(vector<T> a, vector<T> b) {
  vector<T> ret;
  ret = a;
  fora(v, b) ret += v;
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, 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) {
  fora(bv, b) { a[bv.first] += bv.second; }
  return a;
}
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(vector<T> &v, unsigned ll i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(vector<T> &v, unsigned ll s, unsigned ll e) {
  v.erase(v.begin() + s, v.begin() + e);
}
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);
}
template <typename T> void insert(set<T> &v, vector<T> list) {
  for (auto &&va : list)
    v.insert(va);
}
template <class T> T poll(set<T> &S) {
  T ret = *S.begin();
  S.erase(S.begin());
  return ret;
}
template <class T> T poll(mset<T> &S) {
  T ret = *S.begin();
  S.erase(S.begin());
  return ret;
}
template <class T> T poll_back(set<T> &S) {
  T ret = *S.rbegin();
  S.erase(S.rbegin());
  return ret;
}
template <class T> T poll_back(mset<T> &S) {
  T ret = *S.rbegin();
  S.erase(S.rbegin());
  return ret;
}
template <class T> T peek(set<T> &S) {
  T ret = *S.begin();
  return ret;
}
template <class T> T peek(mset<T> &S) {
  T ret = *S.begin();
  return ret;
}
template <class T> T peek_back(set<T> &S) {
  T ret = *S.rbegin();
  return ret;
}
template <class T> T peek_back(mset<T> &S) {
  T ret = *S.rbegin();
  return ret;
}
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); }
void yn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void Yn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void YN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fyn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
  exit(0);
}
void fYn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
  exit(0);
}
void fYN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
  exit(0);
}
void fAb(bool a) {
  if (a)
    cout << "Alice" << endl;
  else
    cout << "Bob";
}
void Possible(bool a) {
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}
void POSSIBLE(bool a) {
  if (a)
    cout << "POSSIBLE" << endl;
  else
    cout << "IMPOSSIBLE" << endl;
  exit(0);
}
void fPossible(bool a) {
  Possible(a);
  exit(0);
}
void fPOSSIBLE(bool a) {
  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 T> vector<vector<T>> cut(vector<T> &a, int tou = 2) {
  int N = sz(a);
  vector<vector<T>> 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 T> vector<vector<T>> cutn(vector<T> &a, int len) {
  int N = sz(a);
  vector<vector<T>> 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;
}
vi inds_(vi &a) {
  int n = sz(a);
  vb was(n);
  vi res(n);
  rep(i, n) {
    assert(!was[a[i]]);
    res[a[i]] = i;
    was[a[i]] = true;
  }
  return res;
}

//@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]
// 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])

// 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
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;
}
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: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;
constexpr int NUM_ = 1e6 + 100;
constexpr int mo = 1e9 + 7;
ll fac[NUM_ + 1], factr[NUM_ + 1], inv[NUM_ + 1];

void init() {
  int i;
  inv[1] = fac[0] = factr[0] = 1;
  for (int i = 2; i <= NUM_; ++i)
    inv[i] = inv[mo % i] * (mo - mo / i) % mo;
  for (int i = 1; i <= NUM_; ++i)
    fac[i] = fac[i - 1] * i % mo, factr[i] = factr[i - 1] * inv[i] % mo;
}
ll com(ll N_, ll C_) {
  if (C_ < 0 || C_ > N_)
    return 0;
  return factr[C_] * fac[N_] % mo * factr[N_ - C_] % mo;
}
int yoko(int N, int B) {
  return (((N + 1) * com(N, B)) % mo * inv[B + 1]) % mo;
  ;
}
void solve() {
  din(H1, W1, H2, W2);
  init();
  auto calc = [&](int H, int W) {
    int res = 0;
    rep(h, H + 1) {
      if (h)
        res += yoko(W + h, h) - yoko(h - 1, h);
      else {
        res += yoko(W + h, h);
      }
      res %= mo;
    }
    return res;
  };
  int res = calc(H2, W2);
  deb(res);
  res -= calc(H1 - 1, W2);
  res %= mo;
  res += mo;
  res %= mo;
  deb(res);
  res -= calc(H2, W1 - 1);
  res %= mo;
  res += mo;
  res %= mo;
  deb(res);
  res += calc(H1 - 1, W1 - 1);
  res %= mo;
  out(res);
}

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) {
    //        cout << "完璧 : solveを書き直そう" << endl;
    //        cout << "     : そして、solve()を呼び出すのだ" << endl;
    //        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
  }
  if (sz(message)) {
    cerr << "****************************" << endl;
    cerr << message << endl;
    cerr << "****************************" << endl;
  }
#endif
  return 0;
};


Fixed submission:
// #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 <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;
#define int long long // todo 消したら動かない intの代わりにsignedを使う
#define ll long long
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

/*@formatter:off*/
//[-n, n)にアクセスできる
// また、外部関数resizeに渡せる
// sizeは[-n, n)でnを返す
template <class T> class mvec {
  vector<T> v;
  int n;

public:
  mvec() : n(0), v(0) {}
  mvec(int n) : n(n), v(n * 2) {}
  mvec(int n, T val) : n(n), v(n * 2, val) {}
  auto &operator[](int i) { return v[i + n]; }
  auto size() { return n; }
  void resize(int sn) {
    assert(n == 0);
    n = sn;
    v.resize(sn * 2);
  }
  auto begin() { return v.begin(); }
  auto rbegin() { return v.rbegin(); }
  auto end() { return v.end(); }
  auto rend() { return v.rend(); }
};
//[]でboolは参照を返さないため特殊化が必要
template <> struct mvec<bool> {
  vector<bool> v;
  int n;
  mvec() : n(0), v(0) {}
  mvec(int n) : n(n), v(n * 2) {}
  mvec(int n, bool val) : n(n), v(n * 2, val) {}
  auto operator[](int i) { return v[i + n]; }
  auto size() { return v.size(); }
  void resize(int sn) {
    assert(n == 0);
    n = sn;
    v.resize(sn * 2);
  }
  auto begin() { return v.begin(); }
  auto rbegin() { return v.rbegin(); }
  auto end() { return v.end(); }
  auto rend() { return v.rend(); }
};
template <class T> ostream &operator<<(ostream &os, mvec<T> &a) {
  int spa = 3;
  for (auto &&v : a) {
    spa = max(spa, (int)(to_string(v).size()) + 1);
  }
  int n = (int)a.size();
  os << endl;
  for (int i = -n; i < n; i++) {
    int need = spa - ((int)to_string(i).size());
    if (i == -n) {
      need -= min(need, spa - ((int)to_string(a[i]).size()));
    }
    while (need--) {
      os << " ";
    }
    os << i;
  }
  os << endl;
  int i = -n;
  for (auto &&v : a) {
    int need = spa - ((int)to_string(v).size());
    if (i == -n) {
      need -= min(need, spa - ((int)to_string(i).size()));
    }
    while (need--) {
      os << " ";
    }
    os << v;
    i++;
  }
  return os;
}
#define mv mvec
#define MV mvec
using mvi = mvec<ll>;
using mvb = mvec<bool>;
using mvs = mvec<string>;
using mvd = mvec<double>;
using mvc = mvec<char>;

#define mvvt0(t) mvec<mvec<t>>
#define mvvt1(t, a) mvec<mvec<t>> a
#define mvvt2(t, a, b) mvec<mvec<t>> a(b)
#define mvvt3(t, a, b, c) mvec<mvec<t>> a(b, mvec<t>(c))
#define mvvt4(t, a, b, c, d) mvec<mvec<t>> a(b, mvec<t>(c, d))
#define mvvi(...)                                                              \
  over4(__VA_ARGS__, mvvt4, mvvt3, mvvt2, mvvt1, mvvt0)(ll, __VA_ARGS__)

template <typename T> mvec<T> make_mv(size_t a) { return mvec<T>(a); }
template <typename T, typename... Ts> auto make_mv(size_t a, Ts... ts) {
  return mvec<decltype(make_mv<T>(ts...))>(a, make_mv<T>(ts...));
}
#define mvni(name, ...) auto name = make_mv<ll>(__VA_ARGS__)

#ifdef _DEBUG
string message;
// 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
  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
// 区間削除は出来ない
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>
// gp_hash_tableでcountを使えないようにするため
template <class T, class U> struct my_unordered_map {
  unordered_map<T, U> m;
  my_unordered_map(){};
  auto begin() { return m.begin(); }
  auto end() { return m.end(); }
  auto cbegin() { return m.cbegin(); }
  auto cend() { return m.cend(); }
  template <class V> auto erase(V v) { return m.erase(v); }
  void clear() {
    m.clear();
  } /*countは gp_hash_tableに存在しない*/ /*!= m.end()*/
  template <class V> auto find(V v) { return m.find(v); }
  template <class V> auto &operator[](V n) { return m[n]; }
};
#define unordered_map my_unordered_map
#define umapi unordered_map<ll, ll>
#define umapp unordered_map<P, ll>
#define umapu unordered_map<uint64_t, ll>
#define umapip unordered_map<ll, P>

#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>

#define umapi __gnu_pbds::gp_hash_table<ll, ll, xorshift>
#define umapp __gnu_pbds::gp_hash_table<P, ll, xorshift>
#define umapu __gnu_pbds::gp_hash_table<uint64_t, ll, xorshift>
#define umapip __gnu_pbds::gp_hash_table<ll, P, xorshift>

#endif
/*@formatter:on*/
struct xorshift {
  static uint64_t splitmix64(uint64_t x) {
    x += 0x9e3779b97f4a7c15;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
  }
  size_t operator()(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()(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);
  }
};
/*@formatter:off*/
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);
}
// 衝突対策
#define ws ws_

template <class A, class B, class C> struct T2 {
  A f;
  B s;
  C t;
  T2() { f = 0, s = 0, t = 0; }
  T2(A f, B s, C t) : f(f), s(s), t(t) {}
  bool operator<(const T2 &r) const {
    return f != r.f   ? f < r.f
           : s != r.s ? s < r.s
                      : t < r.t; /*return f != r.f ? f > r.f : s != r.s ?n s >
                                    r.s : t > r.t; 大きい順 */
  }
  bool operator>(const T2 &r) const {
    return f != r.f   ? f > r.f
           : s != r.s ? s > r.s
                      : t > r.t; /*return f != r.f ? f > r.f : s != r.s ? s >
                                    r.s : t > r.t; 小さい順 */
  }
  bool operator==(const T2 &r) const {
    return f == r.f && s == r.s && t == r.t;
  }
  bool operator!=(const T2 &r) const {
    return f != r.f || s != r.s || t != r.t;
  }
};
template <class A, class B, class C, class D> struct F2 {
  A a;
  B b;
  C c;
  D d;
  F2() { a = 0, b = 0, c = 0, d = 0; }
  F2(A a, B b, C c, D d) : a(a), b(b), c(c), d(d) {}
  bool operator<(const F2 &r) const {
    return a != r.a   ? a < r.a
           : b != r.b ? b < r.b
           : c != r.c ? c < r.c
                      : d < r.d; /*    return a != r.a ? a > r.a : b != r.b ? b
                                    > r.b : c != r.c ? c > r.c : d > r.d;*/
  }
  bool operator>(const F2 &r) const {
    return a != r.a   ? a > r.a
           : b != r.b ? b > r.b
           : c != r.c ? c > r.c
                      : d > r.d; /*        return a != r.a ? a < r.a : b != r.b
                                    ? b < r.b : c != r.c ? c < r.c : d < r.d;*/
  }
  bool operator==(const F2 &r) const {
    return a == r.a && b == r.b && c == r.c && d == r.d;
  }
  bool operator!=(const F2 &r) const {
    return a != r.a || b != r.b || c != r.c || d != r.d;
  }
  ll operator[](ll i) {
    assert(i < 4);
    return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;
  }
};
typedef T2<ll, ll, ll> T;
typedef F2<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); }

//@マクロ省略系 型,構造
#define double long double
using dou = double;
const double eps = 1e-9;
// 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) { 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; }};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 b, epsdou a){return a.v o
(dou)b;} #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(/)
#undef double
#define double epsdou
*/

#define ull unsigned long long
using itn = int;
using str = string;
using bo = bool;
#define au auto
using P = pair<ll, ll>;
using mvp = mvec<P>;
using mvt = mvec<T>;
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()

#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; }

// 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 vvi(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(ll, __VA_ARGS__)
#define vvb(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(bool, __VA_ARGS__)
#define vvs(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(string, __VA_ARGS__)
#define vvd(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(double, __VA_ARGS__)
#define vvc(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(char, __VA_ARGS__)
#define vvp(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(P, __VA_ARGS__)
#define vvt(...)                                                               \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(T, __VA_ARGS__)
#define vv(type, ...)                                                          \
  over4(__VA_ARGS__, vvt4, vvt3, vvt2, vvt1, vvt0)(type, __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
// 任意のマクロサポート用 使う度に初期化する
int index_, v1_, v2_, v3_;

template <class T> struct pq {
  priority_queue<T, vector<T>, greater<T>> q; /*小さい順*/
  T su = 0;
  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> struct pqg {
  priority_queue<T> q; /*大きい順*/
  T su = 0;
  void clear() {
    q = priority_queue<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(); }
};
#define pqi pq<ll>
#define pqgi pqg<ll>
// マクロ 繰り返し
// ↓@オーバーロード隔離

#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)
// 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 I = (int)a.size(), J = (int)a[0].size(), i = 0, j = 0; i < I;       \
       nex_repv2(i, j, I, J, 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 I = (int)a.size(), J = (int)a[0].size(), K = (int)a[0][0].size(),   \
           i = 0, j = 0, k = 0;                                                \
       i < I; nex_repv3(i, j, k, I, J, K, 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]))

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 form1(st)                                                              \
  for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form3(k, v, st)                                                        \
  for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form4(k, v, st, r)                                                     \
  for (auto &&form_it = st.begin(); form_it != st.end() && (*form_it).fi < r;  \
       ++form_it)
#define form5(k, v, st, l, r)                                                  \
  for (auto &&form_it = st.lower_bound(l);                                     \
       form_it != st.end() && (*form_it).fi < r; ++form_it)

#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)

#define fors1(st)                                                              \
  for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors2(v, st)                                                           \
  for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors3(v, st, r)                                                        \
  for (auto &&fors_it = st.begin(); fors_it != st.end() && (*fors_it) < r;     \
       ++fors_it)
#define fors4(v, st, l, r)                                                     \
  for (auto &&fors_it = st.lower_bound(l);                                     \
       fors_it != st.end() && (*fors_it) < r; ++fors_it)

#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)

template <class U> vector<U> to1d(vector<U> &a) { return a; }
template <class U> vector<U> 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 fora_init_2(a, A) ;
#define fora_init_3(fora_i, a, A) auto &&a = A[fora_i];
#define fora_init_4(a, b, A, B)                                                \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];
#define fora_init_5(fora_i, a, b, A, B)                                        \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];
#define fora_init_6(a, b, c, A, B, C)                                          \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];
#define fora_init_7(fora_i, a, b, c, A, B, C)                                  \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];
#define fora_init_8(a, b, c, d, A, B, C, D)                                    \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];                                                        \
  auto &&d = D[fora_i];
#define fora_init_9(fora_i, a, b, c, d, A, B, C, D)                            \
  auto &&a = A[fora_i];                                                        \
  auto &&b = B[fora_i];                                                        \
  auto &&c = C[fora_i];                                                        \
  auto &&d = D[fora_i];
#define fora_init(...)                                                         \
  over9(__VA_ARGS__, fora_init_9, fora_init_8, fora_init_7, fora_init_6,       \
        fora_init_5, fora_init_4, fora_init_3, fora_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_init(k, v, ...)                                                   \
  auto &&k = (*forp_it).fi;                                                    \
  auto &&v = (*forp_it).se;
#define form_init(k, v, ...)                                                   \
  auto &&k = (*form_it).fi;                                                    \
  auto &&v = (*form_it).se;
#define forrm_init(k, v, ...)                                                  \
  auto &&k = (*forrm_it).fi;                                                   \
  auto &&v = (*forrm_it).se;
#define fors_init(v, ...) auto &&v = (*fors_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__);

#define fora_2(a, A) for (auto &&a : A)
#define fora_3(fora_i, a, A) rep(fora_i, sz(A))
#define fora_4(a, b, A, B) rep(fora_i, sz(A))
#define fora_5(fora_i, a, b, A, B) rep(fora_i, sz(A))
#define fora_6(a, b, c, A, B, C) rep(fora_i, sz(A))
#define fora_7(fora_i, a, b, c, A, B, C) rep(fora_i, sz(A))
#define fora_8(a, b, c, d, A, B, C, D) rep(fora_i, sz(A))
#define fora_9(fora_i, a, b, c, d, A, B, C, D) rep(fora_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)

// ↑@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__)
// 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)

#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(...)                                                              \
  over9(__VA_ARGS__, forr_9, forr_8, forr_7, forr_6, forr_5, forr_4, forr_3,   \
        forr_2)(__VA_ARGS__)

// #define forr(v, a) for(auto&& v : makeReversedRange(a))
// 参照を取らない
#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))

#define form(...)                                                              \
  over5(__VA_ARGS__, form5, form4, form3, form2, form1)(__VA_ARGS__)
#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 forp(k, v, st)                                                         \
  for (auto &&forp_it = st.begin(); forp_it != st.end(); ++forp_it)

// マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010
const ll inf = (ll)1e9 + 100;
const ll linf = (ll)1e18 + 100;
const dou dinf = (dou)linf * linf;
const char infc = '{';
const string infs = "{";
const double PI = 3.1415926535897932384626433832795029L;

// 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 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()

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:on*/
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 << p.f << " " << p.s << " " << p.t;
  return os;
}
ostream &operator<<(ostream &os, F p) {
  os << p.a << " " << p.b << " " << p.c << " " << p.d;
  return os;
}
template <typename T> ostream &operator<<(ostream &os, 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, 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, map<T, U> &m) {
  os << endl;
  for (auto &&v : m)
    os << v << endl;
  return os;
}
template <class T> ostream &operator<<(ostream &os, set<T> s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, mset<T> s) {
  fora(v, s) { os << v << " "; }
  return os;
}
template <class T> ostream &operator<<(ostream &os, deque<T> a) {
  fora(v, a) os << v << " ";
  return os;
}
ostream &operator<<(ostream &os, vector<vector<char>> &vec) {
  rep(h, sz(vec)) {
    rep(w, sz(vec[0])) { os << vec[h][w]; }
    os << endl;
  }
  return os;
}
// template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>&
// a) {fora(v,a)os<<v<<endl;return os;}
/*@formatter:off*/
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...);
}
// template<typename W, typename H> void resize(vector<W> &vec, const H head) {
// vec.resize(head); } template<typename W, typename H, typename ... T> void
// resize(vector<W> &vec, const H &head, const T ... tail) {vec.resize(head);for
// (auto &v: vec)resize(v, tail...);}
template <typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool all_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (!all_of2(v[i], f))
      return false;
  }
  return true;
}
template <typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool any_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (any_of2(v[i], f))
      return true;
  }
  return false;
}
template <typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template <typename T, typename F> bool none_of2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (none_of2(v[i], f))
      return false;
  }
  return true;
}
template <typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll find_if2(vector<T> &v, F f) {
  rep(i, sz(v)) {
    if (find_if2(v[i], f))
      return i;
  }
  return sz(v);
}
template <typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll rfind_if2(vector<T> &v, F f) {
  rer(i, sz(v) - 1) {
    if (rfind_if2(v[i], f))
      return i;
  }
  return -1;
}
template <class T> bool contains(string &s, const T &v) {
  return s.find(v) != string::npos;
}
template <typename T> bool contains(vector<T> &v, const T &val) {
  return std::find(v.begin(), v.end(), val) != v.end();
}
template <typename T, typename F> bool contains_if2(vector<T> &v, F f) {
  return find_if(v.begin(), v.end(), f) != v.end();
}
template <typename T, typename F> ll count_if2(T &v, F f) { return f(v); }
template <typename T, typename F> ll count_if2(vector<T> &vec, F f) {
  ll ret = 0;
  fora(v, vec) ret += count_if2(v, f);
  return ret;
}
template <typename T, typename F> void for_each2(T &v, F f) { f(v); }
template <typename T, typename F> void for_each2(vector<T> &vec, F f) {
  fora(v, vec) for_each2(v, f);
}
template <typename W> ll count_od(vector<W> &a) {
  return count_if2(a, [](ll v) { return v & 1; });
}
template <typename W> ll count_ev(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(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(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> l_erase_if2(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> ll l_rfind_if(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(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(...) over3(__VA_ARGS__, 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)

#define for_each_s__2(a, right)                                                \
  do {                                                                         \
    fora(v, a) { v right; }                                                    \
  } while (0)
#define for_each_s__3(a, v, siki)                                              \
  do {                                                                         \
    fora(v, a) { siki; }                                                       \
  } while (0)
#define for_each_s(...)                                                        \
  over3(__VA_ARGS__, for_each_s__3, for_each_s__2)(__VA_ARGS__)

#define for_each__2(a, right) for_each2(a, lamr(right))
#define for_each__3(a, v, siki) for_each2(a, [&](auto v) { return siki; })
#define for_each(...) over3(__VA_ARGS__, for_each__3, for_each__2)(__VA_ARGS__)
#define for_each_f(a, f) for_each2(a, f);

#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)

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());
}
void replace(str &a, char key, char v) { replace(all(a), key, v); }
// 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;
}
// template<class T> void replace(T &a) { replace(a, '#'); }
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(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 includes(vector<T> &a, vector<T> &b) {
  vi c = a;
  vi d = b;
  sort(all(c));
  sort(all(d));
  return includes(all(c), all(d));
}
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));
}
void iota(vector<ll> &ve, ll s, ll n) {
  ve.resize(n);
  iota(all(ve), s);
}
vi iota(ll s, ll len) {
  vi ve(len);
  iota(all(ve), s);
  return ve;
}
template <class A, class B> auto vtop(vector<A> &a, vector<B> &b) {
  assert(sz(a) == sz(b)); /*stringを0で初期化できない  */
  vector<pair<A, B>> res;
  rep(i, sz(a)) res.eb(a[i], b[i]);
  return res;
}
template <class A, class B>
void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {
  a.resize(sz(p)), b.resize(sz(p));
  rep(i, sz(p)) a[i] = p[i].fi, b[i] = p[i].se;
}
template <class A, class B, class C>
auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {
  assert(sz(a) == sz(b) && sz(b) == sz(c));
  vector<T2<A, B, C>> res;
  rep(i, sz(a)) res.eb(a[i], b[i], c[i]);
  return res;
}
template <class A, class B, class C, class D>
auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));
  vector<F2<A, B, C, D>> res;
  rep(i, sz(a)) res.eb(a[i], b[i], c[i], d[i]);
  return res;
}
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {
  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
};
template <class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {
  typedef pair<A, B> U;
  if (type == fisi)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });
  else if (type == fisd)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });
  else if (type == fdsi)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });
  else if (type == fdsd)
    sort(all(a),
         [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });
  else if (type == sifi)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });
  else if (type == sifd)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });
  else if (type == sdfi)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });
  else if (type == sdfd)
    sort(all(a),
         [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
};
template <class U> void sort(vector<U> &a, pcomparator type) {
  if (type == fisi)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });
  else if (type == fisd)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });
  else if (type == fdsi)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });
  else if (type == fdsd)
    sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });
  else if (type == sifi)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });
  else if (type == sifd)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });
  else if (type == sdfi)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });
  else if (type == sdfd)
    sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });
};
template <class A, class B, class C, class D>
void sort(vector<F2<A, B, C, D>> &a, pcomparator type) {
  typedef F2<A, B, C, D> U;
  if (type == fisi)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; });
  else if (type == fisd)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; });
  else if (type == fdsi)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; });
  else if (type == fdsd)
    sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; });
  else if (type == sifi)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; });
  else if (type == sifd)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; });
  else if (type == sdfi)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; });
  else if (type == sdfd)
    sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });
};
template <class U> void sort(vector<U> &a, tcomparator type) {
  if (type == 0)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t;
    });
  else if (type == 1)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t;
    });
  else if (type == 2)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t;
    });
  else if (type == 3)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t;
    });
  else if (type == 4)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t;
    });
  else if (type == 5)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t;
    });
  else if (type == 6)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t;
    });
  else if (type == 7)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t;
    });
  else if (type == 8)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s;
    });
  else if (type == 9)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s;
    });
  else if (type == 10)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s;
    });
  else if (type == 11)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s;
    });
  else if (type == 12)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s;
    });
  else if (type == 13)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s;
    });
  else if (type == 14)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s;
    });
  else if (type == 15)
    sort(all(a), [&](U l, U r) {
      return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s;
    });
  else if (type == 16)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t;
    });
  else if (type == 17)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t;
    });
  else if (type == 18)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t;
    });
  else if (type == 19)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t;
    });
  else if (type == 20)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t;
    });
  else if (type == 21)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t;
    });
  else if (type == 22)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t;
    });
  else if (type == 23)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t;
    });
  else if (type == 24)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f;
    });
  else if (type == 25)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f;
    });
  else if (type == 26)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f;
    });
  else if (type == 27)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f;
    });
  else if (type == 28)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f;
    });
  else if (type == 29)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f;
    });
  else if (type == 30)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f;
    });
  else if (type == 31)
    sort(all(a), [&](U l, U r) {
      return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f;
    });
  else if (type == 32)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s;
    });
  else if (type == 33)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s;
    });
  else if (type == 34)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s;
    });
  else if (type == 35)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s;
    });
  else if (type == 36)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s;
    });
  else if (type == 37)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s;
    });
  else if (type == 38)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s;
    });
  else if (type == 39)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s;
    });
  else if (type == 40)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f;
    });
  else if (type == 41)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f;
    });
  else if (type == 42)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f;
    });
  else if (type == 43)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f;
    });
  else if (type == 44)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f;
    });
  else if (type == 45)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f;
    });
  else if (type == 46)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f;
    });
  else if (type == 47)
    sort(all(a), [&](U l, U r) {
      return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f;
    });
}
template <class A, class B, class C, class D>
void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {
  typedef F2<A, B, C, D> U;
  if (type == 0)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c;
    });
  else if (type == 1)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c;
    });
  else if (type == 2)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c;
    });
  else if (type == 3)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c;
    });
  else if (type == 4)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c;
    });
  else if (type == 5)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c;
    });
  else if (type == 6)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c;
    });
  else if (type == 7)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c;
    });
  else if (type == 8)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b;
    });
  else if (type == 9)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b;
    });
  else if (type == 10)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b;
    });
  else if (type == 11)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b;
    });
  else if (type == 12)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b;
    });
  else if (type == 13)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b;
    });
  else if (type == 14)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b;
    });
  else if (type == 15)
    sort(all(a), [&](U l, U r) {
      return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b;
    });
  else if (type == 16)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c;
    });
  else if (type == 17)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c;
    });
  else if (type == 18)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c;
    });
  else if (type == 19)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c;
    });
  else if (type == 20)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c;
    });
  else if (type == 21)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c;
    });
  else if (type == 22)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c;
    });
  else if (type == 23)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c;
    });
  else if (type == 24)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a;
    });
  else if (type == 25)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a;
    });
  else if (type == 26)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a;
    });
  else if (type == 27)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a;
    });
  else if (type == 28)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a;
    });
  else if (type == 29)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a;
    });
  else if (type == 30)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a;
    });
  else if (type == 31)
    sort(all(a), [&](U l, U r) {
      return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a;
    });
  else if (type == 32)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b;
    });
  else if (type == 33)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b;
    });
  else if (type == 34)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b;
    });
  else if (type == 35)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b;
    });
  else if (type == 36)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b;
    });
  else if (type == 37)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b;
    });
  else if (type == 38)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b;
    });
  else if (type == 39)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b;
    });
  else if (type == 40)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a;
    });
  else if (type == 41)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a;
    });
  else if (type == 42)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a;
    });
  else if (type == 43)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a;
    });
  else if (type == 44)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a;
    });
  else if (type == 45)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a;
    });
  else if (type == 46)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a;
    });
  else if (type == 47)
    sort(all(a), [&](U l, U r) {
      return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a;
    });
}

void sort(string &a) { sort(all(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> void sort(vector<T> &a) { sort(all(a)); }
// P l, P rで f(P) の形で渡す
template <class U, class F> void sort(vector<U> &a, F f) {
  sort(all(a), [&](U l, U r) { return f(l) < f(r); });
};
template <class T> void rsort(vector<T> &a) { sort(all(a), greater<T>()); };
template <class U, class F> void rsort(vector<U> &a, F f) {
  sort(all(a), [&](U l, U r) { return f(l) > f(r); });
};

// F = T<T>
// 例えばreturn p.fi + p.se;
template <class A, class B> void sortp(vector<A> &a, vector<B> &b) {
  auto c = vtop(a, b);
  sort(c);
  rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;
}
template <class A, class B, class F>
void sortp(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  sort(c, f);
  rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;
}
template <class A, class B> void rsortp(vector<A> &a, vector<B> &b) {
  auto c = vtop(a, b);
  rsort(c);
  rep(i, sz(a)) a[i] = c[i].first, b[i] = c[i].second;
}
template <class A, class B, class F>
void rsortp(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  rsort(c, f);
  rep(i, sz(a)) a[i] = c[i].first, b[i] = c[i].second;
}
template <class A, class B, class C>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c) {
  auto d = vtot(a, b, c);
  sort(d);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class F>
void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  sort(d, f);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c) {
  auto d = vtot(a, b, c);
  rsort(d);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class F>
void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  rsort(d, f);
  rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template <class A, class B, class C, class D>
void sortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  auto e = vtof(a, b, c, d);
  sort(e);
  rep(i, sz(a)) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
template <class A, class B, class C, class D>
void rsortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
  auto e = vtof(a, b, c, d);
  rsort(e);
  rep(i, sz(a)) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
// sortindex 元のvectorはソートしない
template <class T> vi sorti(vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(a));
  sortp(b, ind);
  return ind;
} /*indexの分で型が変わるためpcomparatorが必要*/
template <class T> vi sorti(vector<T> &a, pcomparator f) {
  auto b = a;
  vi ind = iota(0, sz(a));
  sortp(b, ind, f);
  return ind;
}
template <class T, class F> vi sorti(vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(a[x]) < f(a[y]); });
  return ind;
}
template <class T> vi rsorti(vector<T> &a) {
  auto b = a;
  vi ind = iota(0, sz(a));
  rsortp(b, ind);
  return ind;
}
template <class T, class F> vi rsorti(vector<T> &a, F f) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(a[x]) > f(a[y]); });
  return ind;
}
template <class A, class B, class F>
vi sortpi(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(c[x]) < f(c[y]); });
  return ind;
}
template <class A, class B>
vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {
  vi ind = iota(0, sz(a));
  auto c = a;
  auto d = b;
  sortt(c, d, ind, f);
  return ind;
}
template <class A, class B> vi sortpi(vector<A> &a, vector<B> &b) {
  return sortpi(a, b, fisi);
};
template <class A, class B, class F>
vi rsortpi(vector<A> &a, vector<B> &b, F f) {
  auto c = vtop(a, b);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(c[x]) > f(c[y]); });
  return ind;
}
template <class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) {
  return sortpi(a, b, fdsd);
};
template <class A, class B, class C, class F>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(d[x]) < f(d[y]); });
  return ind;
}
template <class A, class B, class C>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {
  vi ind = iota(0, sz(a));
  auto d = vtof(a, b, c, ind);
  sort(d, f);
  rep(i, sz(a)) ind[i] = d[i].d;
  return ind;
}
template <class A, class B, class C>
vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) {
    if (a[x] == a[y]) {
      if (b[x] == b[y])
        return c[x] < c[y];
      else
        return b[x] < b[y];
    } else {
      return a[x] < a[y];
    }
  });
  return ind;
}
template <class A, class B, class C, class F>
vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
  auto d = vtot(a, b, c);
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) { return f(d[x]) > f(d[y]); });
  return ind;
}
template <class A, class B, class C>
vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {
  vi ind = iota(0, sz(a));
  sort(all(ind), [&](ll x, ll y) {
    if (a[x] == a[y]) {
      if (b[x] == b[y])
        return c[x] > c[y];
      else
        return b[x] > b[y];
    } else {
      return a[x] > a[y];
    }
  });
  return ind;
}
template <class T> void sort2(vector<vector<T>> &a) {
  for (ll i = 0, n = a.size(); i < n; ++i)
    sort(a[i]);
}
template <class T> void rsort2(vector<vector<T>> &a) {
  for (ll i = 0, n = a.size(); i < n; ++i)
    rsort(a[i]);
}

template <class... T, class U> auto sorted(U head, T... a) {
  sort(head, a...);
  return head;
}

template <typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) {
  rep(i, N) a[i] = v;
}
template <typename A, size_t N, size_t O, typename T>
void fill(A (&a)[N][O], const T &v) {
  rep(i, N) rep(j, O) a[i][j] = v;
}
template <typename A, size_t N, size_t O, size_t P, typename T>
void fill(A (&a)[N][O][P], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) a[i][j][k] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, typename T>
void fill(A (&a)[N][O][P][Q], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) a[i][j][k][l] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          typename T>
void fill(A (&a)[N][O][P][Q][R], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) a[i][j][k][l][m] = v;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S, typename T>
void fill(A (&a)[N][O][P][Q][R][S], const T &v) {
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S)
      a[i][j][k][l][m][n] = v;
}
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, ll len, const T v) {
  rep(i, len) xx[i] = v;
}
template <typename W, typename T>
void fill(vector<vector<W>> &xx, int sh, int th, int sw, int tw, T v) {
  rep(h, sh, th) rep(w, sw, tw) xx[h][w] = v;
}
template <class T, class U> void fill(vector<T> &a, vi &ind, U val) {
  fora(v, ind) a[v] = val;
}
template <class W, class T> void fill(mvec<W> &xx, const T v) {
  fora(x, xx) fill(x, v);
}

template <typename A, size_t N> A sum(A (&a)[N]) {
  A res = 0;
  rep(i, N) res += a[i];
  return res;
}
template <typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {
  A res = 0;
  rep(i, N) rep(j, O) res += a[i][j];
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) res += a[i][j][k];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A sum(A (&a)[N][O][P][Q]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) res += a[i][j][k][l];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A sum(A (&a)[N][O][P][Q][R]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) res += a[i][j][k][l][m];
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A sum(A (&a)[N][O][P][Q][R][S]) {
  A res = 0;
  rep(i, N) rep(j, O) rep(k, P) rep(l, Q) rep(m, R) rep(n, S) res +=
      a[i][j][k][l][m][n];
  return res;
}
//@汎用便利関数 入力
ll in() {
  ll 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...);
}

#define din1(a)                                                                \
  ll a;                                                                        \
  cin >> a
#define din2(a, b)                                                             \
  ll a, b;                                                                     \
  cin >> a >> b
#define din3(a, b, c)                                                          \
  ll a, b, c;                                                                  \
  cin >> a >> b >> c
#define din4(a, b, c, d)                                                       \
  ll a, b, c, d;                                                               \
  cin >> a >> b >> c >> d
#define din5(a, b, c, d, e)                                                    \
  ll a, b, c, d, e;                                                            \
  cin >> a >> b >> c >> d >> e
#define din6(a, b, c, d, e, f)                                                 \
  ll a, b, c, d, e, f;                                                         \
  cin >> a >> b >> c >> d >> e >> f
#define din(...)                                                               \
  over6(__VA_ARGS__, din6, din5, din4, din3, din2, din1)(__VA_ARGS__)

#define dins1(a)                                                               \
  str a;                                                                       \
  cin >> a
#define dins2(a, b)                                                            \
  str a, b;                                                                    \
  cin >> a >> b
#define dins3(a, b, c)                                                         \
  str a, b, c;                                                                 \
  cin >> a >> b >> c
#define dins4(a, b, c, d)                                                      \
  str a, b, c, d;                                                              \
  cin >> a >> b >> c >> d
#define dins5(a, b, c, d, e)                                                   \
  str a, b, c, d, e;                                                           \
  cin >> a >> b >> c >> d >> e
#define dins6(a, b, c, d, e, f)                                                \
  str a, b, c, d, e, f;                                                        \
  cin >> a >> b >> c >> d >> e >> f
#define dins(...)                                                              \
  over6(__VA_ARGS__, dins6, dins5, dins4, dins3, dins2, dins1)(__VA_ARGS__)

#define din1d(a)                                                               \
  din1(a);                                                                     \
  a--
#define din2d(a, b)                                                            \
  din2(a, b);                                                                  \
  a--, b--
#define din3d(a, b, c)                                                         \
  din3(a, b, c);                                                               \
  a--, b--, c--
#define din4d(a, b, c, d)                                                      \
  din4(a, b, c, d);                                                            \
  a--, b--, c--, d--
#define dind(...) over4(__VA_ARGS__, din4d, din3d, din2d, din1d)(__VA_ARGS__)

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; }

#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; }
template <class T> string out_m2(vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), 12ll);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) {
    ss << a[i];
    if (typeid(a[i]) == typeid(P)) {
      ss << endl;
    } else {
      ss << " ";
    }
  }
  return ss.str();
}
template <class T>
string out_m2(vector<vector<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  H = min({H, sz(a), 12ll});
  W = min({W, sz(a[0]), 12ll});
  stringstream ss;
  ss << endl;
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  rep(w, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 4 + 3) ss << "_";
  ss << "" << endl;
  rep(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    rep(w, min(sz(a[h]), 12ll)) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  return ss.str();
}
template <class T>
string out_m2(vector<vector<vector<T>>> &a, ll H = inf, ll W = inf,
              ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 12;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << out_m2(a[i], W, U, i);
  }
  return ss.str();
}
template <class T, size_t N> string out_m2(T (&a)[N]) {
  vector<T> b;
  resize(b, N);
  rep(i, N) { b[i] = a[i]; }
  return out_m2(b);
}
template <class T, size_t N, size_t M> string out_m2(T (&a)[N][M]) {
  vector<vector<T>> b;
  resize(b, N, M);
  rep(i, N) {
    rep(j, M) { b[i][j] = a[i][j]; }
  }
  return out_m2(b);
}
template <class T, size_t N, size_t M, size_t O>
string out_m2(T (&a)[N][M][O]) {
  vector<vector<vector<T>>> b;
  resize(b, N, M, O);
  rep(i, N) {
    rep(j, M) {
      rep(k, O) { b[i][j][k] = a[i][j][k]; }
    }
  }
  return out_m2(b);
}
string out_m2(int a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
/*@formatter:on*/
template <class T>
string out_m2(mvec<mvec<T>> &a, ll H = inf, ll W = inf, int key = inf) {
  H = min({H, sz(a), 6ll});
  W = min({W, sz(a[0]), 6ll});
  stringstream ss;
  ss << endl;
  //    if (key == inf)ss << " *|"; else ss << " " << key << "|";
  if (key == inf)
    ss << " *|";
  else {
    ss << std::right << std::setw(2) << key;
    ss << "|";
  }
  rep(w, -W, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 8 + 3) ss << "_";
  ss << "" << endl;
  rep(h, -H, H) {
    ss << std::right << std::setw(2) << h << "|";
    int NW = min(sz(a[h]), 6ll);
    rep(w, -NW, NW) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  return ss.str();
} /*@formatter:on*/
template <class T>
string out_m2(mvec<mvec<mvec<T>>> &a, ll H = inf, ll W = inf, ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 6;
  H = min(H, sz(a));
  rep(i, -H, H) {
    ss << endl;
    ss << out_m2(a[i], W, U, i);
  }
  return ss.str();
}

/*@formatter:off*/
template <class T> string out_m2(T &a) {
  stringstream ss;
  ss << a;
  return ss.str();
}
/*@formatter:on*/
template <class T> string out_m(vector<T> &a, ll W = inf) {
  stringstream ss;
  if (W == inf)
    W = min(sz(a), 12ll);
  if (sz(a) == 0)
    return ss.str();
  rep(i, W) { ss << a[i] << " "; }
  ss << "" << endl;
  return ss.str();
}
/*@formatter:off*/
template <class T>
string out_m(vector<vector<T>> &a, ll H = inf, ll W = inf, int key = -1) {
  H = min({H, sz(a), 12ll});
  W = min({W, sz(a[0]), 12ll});
  stringstream ss;
  ss << endl;
  if (key == -1)
    ss << " *|";
  else
    ss << " " << key << "|";
  rep(w, W) ss << std::right << std::setw(4) << w;
  ss << "" << endl;
  rep(w, W * 4 + 3) ss << "_";
  ss << "" << endl;
  rep(h, H) {
    ss << std::right << std::setw(2) << h << "|";
    rep(w, min(sz(a[h]), 12ll)) {
      if (abs(a[h][w]) == linf)
        ss << "   e"
           << "";
      else
        ss << std::right << std::setw(4) << a[h][w];
    }
    ss << "" << endl;
  }
  ss << endl;
  return ss.str();
}
template <class T>
string out_m(vector<vector<vector<T>>> &a, ll H = inf, ll W = inf, ll U = inf) {
  stringstream ss;
  if (H == inf)
    H = 5;
  H = min(H, sz(a));
  rep(i, H) {
    ss << endl;
    ss << out_m(a[i], W, U, i);
  }
  ss << endl;
  return ss.str();
}
string out_m(int a) {
  stringstream ss;
  ss << a << endl;
  return ss.str();
}
template <class T> string out_m(T &a) {
  stringstream ss;
  ss << a << endl;
  return ss.str();
}
template <class T> void outv(vector<T> &a, ll W = inf) {
  cout << out_m(a, W) << endl;
}
template <class T>
void outv(vector<vector<T>> &a, ll H = linf, ll W = linf, int key = -1) {
  cout << out_m(a, H, W, key) << endl;
}
template <class T>
void outv(vector<vector<vector<T>>> &a, ll H = linf, ll W = linf, ll U = linf) {
  cout << out_m(a, H, W, U) << endl;
}
#else
template <class T> void outv(vector<T> &a, ll W = inf) {
  rep(i, min(W, sz(a))) { cout << a[i] << " "; }
  cout << "" << endl;
}
template <class T>
void outv(vector<vector<T>> &a, ll H = linf, ll W = linf, int key = -1) {
  rep(i, min(H, sz(a))) { outv(a[i], W); }
}
template <class T>
void outv(vector<vector<vector<T>>> &a, ll H = linf, ll W = linf, ll U = linf) {
  ;
}
#define err(...) ;
#endif
template <class T> void outl(vector<T> &a, int n = inf) {
  rep(i, min(n, sz(a))) cout << a[i] << endl;
}
// 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; }
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(n);                                                                     \
  rep(dnai, n) cin >> a[dnai];
#define dnad(a, n)                                                             \
  vi a(n);                                                                     \
  rep(dnai, n) cin >> a[dnai], a[dnai]--;
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(n), b(n);                                                               \
  rep(dna2i, n) cin >> a[dna2i] >> b[dna2i];
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 dna2d(a, b, n)                                                         \
  vi a(n), b(n);                                                               \
  rep(dna2di, n) {                                                             \
    cin >> a[dna2di] >> b[dna2di];                                             \
    a[dna2di]--, b[dna2di]--;                                                  \
  }
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]--;
}
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(n), b(n), c(n);                                                         \
  rep(dna3i, n) cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
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(n), b(n), c(n);                                                         \
  rep(dna3di, n) {                                                             \
    cin >> a[dna3di] >> b[dna3di] >> c[dna3di];                                \
    a[dna3di]--, b[dna3di]--, c[dna3di]--;                                     \
  }
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(n), b(n), c(n), d(n);                                                   \
  rep(dna4i, n) cin >> a[dna4i] >> b[dna4i] >> c[dna4i] >> d[dna4i];
#define dna4d(a, b, c, d, n)                                                   \
  vi a(n), b(n), c(n), d(n);                                                   \
  rep(dna4i, n) cin >> a[dna4i] >> b[dna4i] >> c[dna4i] >> d[dna4i],           \
      --a[dna4i], --b[dna4i], --c[dna4i], --d[dna4i];
#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];
// デバッグ
#define sp << " " <<

#define deb1(x) debugName(x) << " = " << out_m2(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__)

#ifdef _DEBUG
#define deb(...)                                                               \
  cerr << over10(__VA_ARGS__, deb10, deb9, deb8, deb7, deb6, deb5, deb_4,      \
                 deb_3, deb_2, deb1)(__VA_ARGS__)                              \
       << endl

#define base_keta 8
void print_n_base(int x, int base) { cerr << bitset<base_keta>(x) << 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)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 2);
#define deb3(x)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 3);
#define deb4(x)                                                                \
  cerr << debugName(x) << " = ";                                               \
  print_n_base(x, 4);

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

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

//@formatter:off
// よく使うクラス、構造体
// graphでredefineしているため、書き換えたら向こうも書き換える
struct unionfind {
  vector<ll> par;
  vector<ll> siz;
  vector<ll> es;
  ll n, trees; // 連結グループの数(親の種類)
  unionfind(ll n) : n(n), trees(n) {
    par.resize(n);
    siz.resize(n);
    es.resize(n);
    for (ll i = 0; i < n; i++) {
      par[i] = i;
      siz[i] = 1;
    }
  }
  ll root(ll x) {
    if (par[x] == x) {
      return x;
    } else {
      return par[x] = root(par[x]);
    }
  }
  ll operator()(ll x) { return root(x); }
  bool unite(ll x, ll y) {
    x = root(x);
    y = root(y);
    es[x]++;
    if (x == y)
      return false;
    if (siz[x] > siz[y])
      swap(x, y);
    trees--;
    par[x] = y;
    siz[y] += siz[x];
    es[y] += es[x];
    return true;
  }
  bool same(ll x, ll y) { return root(x) == root(y); }
  ll size(ll x) { return siz[root(x)]; }
  ll esize(ll x) { return es[root(x)]; }
  vi sizes() {
    vi cou(n);
    vi ret;
    ret.reserve(n);
    rep(i, n) { cou[root(i)]++; }
    rep(i, n) {
      if (cou[i])
        ret.push_back(cou[i]);
    }
    return ret;
  }
  // つながりを無向グラフと見なし、xが閉路に含まれるか判定
  bool close(ll x) { return esize(x) >= size(x); }
  vector<vi> sets() {
    vi ind(n, -1);
    ll i = 0;
    vvi(res, trees);
    rep(j, n) {
      ll r = root(j);
      if (ind[r] == -1)
        ind[r] = i++;
      res[ind[r]].push_back(j);
    }
    rep(i, trees) {
      ll r = root(res[i][0]);
      if (res[i][0] == r)
        continue;
      rep(j, 1, sz(res[i])) {
        if (res[i][j] == r) {
          swap(res[i][0], res[i][j]);
          break;
        }
      }
    }
    return res;
  }
}; //@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 < 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';
  }
}
bint max(bint a, signed b) { return max(a, (bint)b); }
bint max(signed a, bint b) { return max((bint)a, b); }
bint max(bint a, ll b) { return max(a, (bint)b); }
bint max(ll a, bint b) { return max((bint)a, b); }
bint min(bint a, signed b) { return min(a, (bint)b); }
bint min(signed a, bint b) { return min((bint)a, b); }
bint min(bint a, ll b) { return min(a, (bint)b); }
bint min(ll a, bint b) { return min((bint)a, b); }
// エラー
void ole() {
#ifdef _DEBUG
  debugline("ole");
  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() {
  assert(0 == 1);
  exit(0);
}
void tle() {
  while (inf)
    cout << inf << endl;
}

// 便利関数

// 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;
}
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

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 T> vector<pair<T, int>> run_length(vector<T> &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;
}
vector<pair<char, ll>> run_length(string &a) {
  vector<pair<char, ll>> 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:on*/
template <class T, class F> T mgr(T ok, T ng, F f) {
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
template <class T, class F> T mgr(signed ok_, T ng, F f) {
  T ok = ok_;
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
template <class T, class F> T mgr(T ok, signed ng_, F f) {
  T ng = ng_;
  bool han = true;
  if (ok < ng)
    while (ng - ok > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    while (ok - ng > 1) {
      T mid = (ok + ng) >> 1;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
/*@formatter:off*/
template <class F> dou mgrd(dou ok, dou ng, F f, int kai = 100) {
  bool han = true;
  if (ok < ng)
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  else
    rep(i, kai) {
      dou mid = (ok + ng) / 2;
      if (f(mid))
        ok = mid, han = true;
      else
        ng = mid, han = false;
      deb(mid, han);
    }
  return ok;
}
// 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-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 -1;
}
template <typename W, typename T> ll find(vector<W> &a, const T key) {
  rep(i, sz(a)) if (a[i] == key) return i;
  return -1;
}
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>
T 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 mt(i, j, k);
  return mt(-1, -1, -1);
}
// stringも書く
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 -1;
}
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 -1;
}
int find(string &s, const char key) {
  rep(i, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return -1;
}
int find(string &s, int l, const char key) {
  rep(i, l, sz(s)) {
    if (s[i] == key)
      return i;
  }
  return -1;
}
template <typename W, typename T> ll count2(W &a, const T k) { return a == k; }
template <typename W, typename T> ll count2(vector<W> &a, const T k) {
  ll ret = 0;
  fora(v, a) ret += count2(v, k);
  return ret;
}
template <typename W, typename T> ll count(vector<W> &a, const T k) {
  ll ret = 0;
  fora(v, a) ret += count2(v, k);
  return ret;
}
vi count(vi &a) {
  int ma = 0;
  fora(v, a) {
    if (ma < v)
      ma = v;
  }
  vi res(ma + 1);
  fora(v, a) { res[v]++; }
  return res;
}
ll count(str &a, 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;
}
//'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;
}
vi count(str &a, int r = inf) { return count(a, 0, min(r, sz(a))); }
#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> void rev(vector<T> &a) { reverse(all(a)); }
template <class U> void 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];
  a = b;
}
void rev(string &a) { reverse(all(a)); }
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) {
  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;
        }
      }
    }
  }
}
ll getr(ll a, ll keta) { return (a / (ll)pow(10, keta)) % 10; }
// 上から何桁目か
ll getl(ll a, ll ket) {
  int sketa = keta(a);
  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
//[v] := iとなるようなvectorを返す
// 存在しない物は-1
template <class T> auto keys(T &a) {
  vector<decltype((a.begin())->fi)> res;
  for (auto &&k : a)
    res.push_back(k.fi);
  return res;
}
template <class T> auto values(T &a) {
  vector<decltype((a.begin())->se)> res;
  for (auto &&k : a)
    res.push_back(k.se);
  return res;
}
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;
}
template <class T> constexpr T min(T a, signed b) { return a < b ? a : b; }
template <class T> constexpr T max(T a, signed b) { return a < b ? b : a; }
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;
}
template <class T> T min(vector<T> &a) { return *min_element(all(a)); }
template <class T> T mini(vector<T> &a) {
  return min_element(all(a)) - a.begin();
}
template <class T> T min(vector<T> &a, ll n) {
  return *min_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> T min(vector<T> &a, ll s, ll n) {
  return *min_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <class T> T max(vector<T> &a) { return *max_element(all(a)); }
template <class T, class U> T max(vector<T> &a, vector<U> &b) {
  return max(*max_element(all(a)), *max_element(all(b)));
}
template <class T> T maxi(vector<T> &a) {
  return max_element(all(a)) - a.begin();
}
template <class T> T max(vector<T> &a, ll n) {
  return *max_element(a.begin(), a.begin() + min(n, sz(a)));
}
template <class T> T max(vector<T> &a, ll s, ll n) {
  return *max_element(a.begin() + s, a.begin() + min(n, sz(a)));
}
template <typename A, size_t N> A max(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = max(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> A max(A (&a)[N][O]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A max(A (&a)[N][O][P]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A max(A (&a)[N][O][P][Q], const T &v) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A max(A (&a)[N][O][P][Q][R]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A max(A (&a)[N][O][P][Q][R][S]) {
  A res = max(a[0]);
  rep(i, N) res = max(res, max(a[i]));
  return res;
}
template <typename A, size_t N> A min(A (&a)[N]) {
  A res = a[0];
  rep(i, N) res = min(res, a[i]);
  return res;
}
template <typename A, size_t N, size_t O> A min(A (&a)[N][O]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P> A min(A (&a)[N][O][P]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q>
A min(A (&a)[N][O][P][Q], const T &v) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>
A min(A (&a)[N][O][P][Q][R]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <typename A, size_t N, size_t O, size_t P, size_t Q, size_t R,
          size_t S>
A min(A (&a)[N][O][P][Q][R][S]) {
  A res = min(a[0]);
  rep(i, N) res = min(res, min(a[i]));
  return res;
}
template <class T> T sum(vector<T> &v, ll s, ll t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += v[i];
  return ret;
}
template <class T> T sum(vector<T> &v, ll t = inf) { return sum(v, 0, t); }
template <class T> T sum(vector<vector<T>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<T>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T> T sum(vector<vector<vector<T>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<vector<T>>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T> T sum(vector<vector<vector<vector<T>>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T> T sum(vector<vector<vector<vector<T>>>> &v, int t = inf) {
  return sum(v, 0, t);
}
template <class T>
T sum(vector<vector<vector<vector<vector<T>>>>> &v, int s, int t) {
  T ret = 0;
  rep(i, s, min(sz(v), t)) ret += sum(v[i]);
  return ret;
}
template <class T>
T sum(vector<vector<vector<vector<vector<T>>>>> &v, int t = inf) {
  return sum(v, 0, t);
}

template <class T> auto sum(priority_queue<T, vector<T>, greater<T>> &r) {
  auto q = r;
  T ret = 0;
  while (sz(q)) {
    ret += q.top();
    q.pop();
  }
  return ret;
}
template <class T> auto sum(priority_queue<T> &r) {
  auto q = r;
  T ret = 0;
  while (sz(q)) {
    ret += q.top();
    q.pop();
  }
  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;}
vi v_i(vi &a) {
  int n = max(a) + 1;
  vi ret(n, -1);
  rep(i, sz(a)) { 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;
}
// imoは0-indexed
// ruiは1-indexed
template <class T> vector<T> imo(vector<T> &v) {
  vector<T> ret = v;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
// kと同じものの数
template <class T, class U> vi imo(vector<T> &a, U k) {
  vector<T> ret = a;
  rep(i, sz(ret)) ret[i] = a[i] == k;
  rep(i, sz(ret) - 1) ret[i + 1] += ret[i];
  return ret;
}
template <class T> vector<T> imox(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(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(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(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(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 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) { 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]; }
  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> 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<T> ri;
  bool build = false;

public:
  int n;
  ruimax(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    int ma = MIN(T);
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = -linf;
    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 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;
    }
  }
  T geti(int r = inf) { return geti(0, min(r, n)); };
  T getv(int r = inf) { return getv(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<T> ri;
  bool build = false;
  int n;

public:
  ruimin(vector<T> &a) : ve(a), n(sz(a)) {
    int index = -1;
    int mi = MAX(T);
    rv.resize(n + 1);
    ri.resize(n + 1);
    rv[0] = linf;
    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 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;
    }
  }
  T geti(int r = inf) { return geti(0, min(r, n)); };
  T getv(int r = inf) { return getv(0, min(r, n)); };
  auto begin() { return rv.begin(); }
  auto end() { return rv.end(); }
};

template <class T> ostream &operator<<(ostream &os, ruiC<T> a) {
  fora(v, a.rui) os << v << " ";
  return os;
}
template <class T> vector<T> ruiv(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() { return ruiC<T>(); }
template <class T> ruiC<T> ruic(vector<T> &a) {
  vector<T> ret = ruiv(a);
  return ruiC<T>(ret);
}
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;
}
vector<ll> ruiv(string &a) {
  if (sz(a) == 0)
    return vi(1);
  ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;
  vector<ll> ret(a.size() + 1);
  rep(i, a.size()) ret[i + 1] = ret[i] + a[i] - dec;
  return ret;
}
ruiC<ll> ruic(string &a) {
  vector<ll> ret = ruiv(a);
  return ruiC<ll>(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;
  ruiC2(vector<vector<T>> &ru) : rui(sz(ru)), H(sz(ru)) {
    for (int h = 0; h < H; 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) {
    assert(h < H);
    return rui[h];
  }
  //    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();}
};

// a~zを0~25として
//  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
ruiC2<ll> ruicou(str &a) {
  str s = a;
  replace(s);
  vector<ruiC<ll>> res(26);
  vvi(cou, 26, sz(s));
  rep(i, sz(s)) { cou[s[i]][i] = 1; }
  return ruiC2<ll>(cou);
}
ruiC2<ll> ruicou(vi &a) {
  int H = max(a) + 1;
  vector<ruiC<ll>> res(H);
  vvi(cou, H, sz(a));
  rep(i, 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);
}
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;
}

#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);
}
/*@formatter:off*/
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> 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 ins(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 ins(0, i, sz(a)) && ins(0, j, sz(a));
}
#define inside ins
ll u(ll a) { return a < 0 ? 0 : a; }
template <class T> vector<T> u(vector<T> &a) {
  vector<T> ret = a;
  fora(v, ret) v = u(v);
  return ret;
}

// 添え字を返す
template <class F> ll 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 resIndex;
}
template <class F> ll 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 resIndex;
}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template <class F> dou 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 left;
}
template <class F> dou 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 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> ll 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 mini;
}
template <class F> ll 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 maxi;
}
template <class F>
dou 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 mini;
}
template <class F>
dou 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 maxi;
}
template <class T> T min(vector<vector<T>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, *min_element(all(a[i])));
  return res;
}
template <class T> T max(vector<vector<T>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, *max_element(all(a[i])));
  return res;
}
template <class T> T min(vector<vector<vector<T>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<T>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<T>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<T>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<vector<T>>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<vector<T>>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
template <class T> T min(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {
  T res = MAX(T);
  rep(i, a.size()) chmi(res, min(a[i]));
  return res;
}
template <class T> T max(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {
  T res = MIN(T);
  rep(i, a.size()) chma(res, max(a[i]));
  return res;
}
// pow周りの仕様
// powiを使うと整数型
// powbを使うとbint型
// powを使うと powlに変換され long doubleが返る
#ifdef _DEBUG
auto pow(ll a, ll k) {
  static bool was = 1;
  if (was) {
    message += "if integer use *powi* it's very fast\n";
  }
  was = 0;
  return powl(a, k);
}
// 上のメッセージを出すための関数
auto pow(signed a, ll k) { return pow((ll)a, k); }
auto pow(signed a, signed k) { return pow((ll)a, (ll)k); }
#else
#define pow powl

#endif
// 整数型のpow
int powi(int a, int k) {
  if (a == 2)
    return 1ll << k;
  int res = 1;
  int x = a;
  while (k) {
    if (k & 1)
      res *= x;
    x *= x;
    k >>= 1;
  }
  return res;
}
// define pow powlより上に動かすとバグる
bint pow(bint a, ll k) {
  bint res = 1;
  bint x = a;
  while (k) {
    if (k & 1)
      res *= x;
    x *= x;
    k >>= 1;
  }
  return res;
}
bint pow(bint a, signed k) { return pow(a, (ll)k); }
bint powb(int a, int b) { return pow((bint)a, b); }
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;
}
ll bget(ll m, ll keta, ll sinsuu) {
  m /= (ll)pow(sinsuu, keta);
  return m % sinsuu;
}
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));
}
ll bit(ll n, ll sinsuu) { return (ll)pow(sinsuu, n); }
ll mask(ll n) { return (1ll << n) - 1; }
// aをbitに置きなおす
//{0, 2} -> 101
ll bit(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 bitb(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) { return n & -n; }
ll lbiti(ll n) { return log2(n & -n); }
// 最上位ビット
ll hbit(ll 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) { 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;
}
// masの立ってるindexを見る
#define forbit(i, mas)                                                         \
  for (int forbitj = lbit(mas), forbitm = mas, i = log2(forbitj); forbitm;     \
       forbitm = forbitj ? forbitm ^ forbitj : 0, forbitj = lbit(forbitm),     \
           i = log2(forbitj))

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;
  }
  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 unique(v) v.erase(unique(v.begin(), v.end()), v.end());
//[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;
}
vi compress(vi &a, umap<ll, ll> &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();
  }
}
// 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 lowerBound(ruiC<T> &a, U v, W banpei) {
  return lowerBound(a.rui, v, banpei);
}
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 upperBound(ruiC<T> &a, U v, W banpei) {
  return upperBound(a.rui, v, banpei);
}
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 rlowerBound(ruiC<T> &a, U v, W banpei) {
  return rlowerBound(a.rui, v, banpei);
}
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);
  }
}
template <class T, class U, class W> T rupperBound(ruiC<T> &a, U v, W banpei) {
  return rupperBound(a.rui, v, banpei);
}
#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; };
ll fact(ll v) { return v <= 1 ? 1 : v * fact(v - 1); }
dou factd(int v) {
  static vd fact(2, 1);
  if (sz(fact) <= v) {
    rep(i, sz(fact), v + 1) { fact.push_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];
}
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];
}

ll gcd(ll a, ll b) {
  while (b)
    a %= b, swap(a, b);
  return abs(a);
}
ll gcd(vi b) {
  ll res = b[0];
  rep(i, 1, sz(b)) res = gcd(b[i], res);
  return res;
}
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;
  }
}

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); }

// 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;
}

constexpr int bsetlen = k5 * 2;
// 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> 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, class U> void operator+=(pair<T, U> &a, pair<T, U> &b) {
  a.fi += b.fi;
  a.se += b.se;
}
template <class T, class U> pair<T, U> operator+(pair<T, U> &a, pair<T, U> &b) {
  return pair<T, U>(a.fi + b.fi, a.se + b.se);
}

template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  basic_string<CharT, Traits, Alloc> str(lhs);
  str.append(to_string(rv));
  return str;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
  lhs += to_string(rv);
}
template <typename CharT, typename Traits, typename Alloc>
basic_string<CharT, Traits, Alloc>
operator+(const basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  basic_string<CharT, Traits, Alloc> str(lhs);
  str.append(to_string(rv));
  return str;
}
template <typename CharT, typename Traits, typename Alloc>
void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {
  lhs += to_string(rv);
}
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, U v) { a.push(v); }
template <class T, class U> void operator+=(deque<T> &a, 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, U v) {
  a.push(v);
  return a;
}
template <class T, class U>
priority_queue<T> &operator+=(priority_queue<T> &a, U v) {
  a.push(v);
  return a;
}
template <class T> set<T> &operator+=(set<T> &a, vector<T> v) {
  fora(d, v) a.insert(d);
  return a;
}
template <class T, class U> auto operator+=(set<T> &a, U v) {
  return a.insert(v);
}
template <class T, class U> auto operator-=(set<T> &a, U v) {
  return a.erase(v);
}
template <class T, class U> auto operator+=(mset<T> &a, U v) {
  return a.insert(v);
}
template <class T, class U>
set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {
  a.insert(v);
  return a;
}
template <class T, class U> vector<T> &operator+=(vector<T> &a, U v) {
  a.push_back(v);
  return a;
}
template <class T, class U> vector<T> operator+(vector<T> &a, U v) {
  vector<T> ret = a;
  ret += v;
  return ret;
}
template <class T, class U> vector<T> operator+(U v, vector<T> &a) {
  vector<T> ret = a;
  ret.insert(ret.begin(), v);
  return ret;
}
template <class T> vector<T> operator+(vector<T> a, vector<T> b) {
  vector<T> ret;
  ret = a;
  fora(v, b) ret += v;
  return ret;
}
template <class T> vector<T> &operator+=(vector<T> &a, 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) {
  fora(bv, b) { a[bv.first] += bv.second; }
  return a;
}
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(vector<T> &v, unsigned ll i) {
  v.erase(v.begin() + i);
}
template <typename T> void erase(vector<T> &v, unsigned ll s, unsigned ll e) {
  v.erase(v.begin() + s, v.begin() + e);
}
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);
}
template <typename T> void insert(set<T> &v, vector<T> list) {
  for (auto &&va : list)
    v.insert(va);
}
template <class T> T poll(set<T> &S) {
  T ret = *S.begin();
  S.erase(S.begin());
  return ret;
}
template <class T> T poll(mset<T> &S) {
  T ret = *S.begin();
  S.erase(S.begin());
  return ret;
}
template <class T> T poll_back(set<T> &S) {
  T ret = *S.rbegin();
  S.erase(S.rbegin());
  return ret;
}
template <class T> T poll_back(mset<T> &S) {
  T ret = *S.rbegin();
  S.erase(S.rbegin());
  return ret;
}
template <class T> T peek(set<T> &S) {
  T ret = *S.begin();
  return ret;
}
template <class T> T peek(mset<T> &S) {
  T ret = *S.begin();
  return ret;
}
template <class T> T peek_back(set<T> &S) {
  T ret = *S.rbegin();
  return ret;
}
template <class T> T peek_back(mset<T> &S) {
  T ret = *S.rbegin();
  return ret;
}
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); }
void yn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
}
void Yn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
}
void YN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
}
void fyn(bool a) {
  if (a)
    cout << "yes" << endl;
  else
    cout << "no" << endl;
  exit(0);
}
void fYn(bool a) {
  if (a)
    cout << "Yes" << endl;
  else
    cout << "No" << endl;
  exit(0);
}
void fYN(bool a) {
  if (a)
    cout << "YES" << endl;
  else
    cout << "NO" << endl;
  exit(0);
}
void fAb(bool a) {
  if (a)
    cout << "Alice" << endl;
  else
    cout << "Bob";
}
void Possible(bool a) {
  if (a)
    cout << "Possible" << endl;
  else
    cout << "Impossible" << endl;
  exit(0);
}
void POSSIBLE(bool a) {
  if (a)
    cout << "POSSIBLE" << endl;
  else
    cout << "IMPOSSIBLE" << endl;
  exit(0);
}
void fPossible(bool a) {
  Possible(a);
  exit(0);
}
void fPOSSIBLE(bool a) {
  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 T> vector<vector<T>> cut(vector<T> &a, int tou = 2) {
  int N = sz(a);
  vector<vector<T>> 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 T> vector<vector<T>> cutn(vector<T> &a, int len) {
  int N = sz(a);
  vector<vector<T>> 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;
}
vi inds_(vi &a) {
  int n = sz(a);
  vb was(n);
  vi res(n);
  rep(i, n) {
    assert(!was[a[i]]);
    res[a[i]] = i;
    was[a[i]] = true;
  }
  return res;
}

//@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]
// 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])

// 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
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;
}
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: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;
constexpr int NUM_ = 2e6 + 100;
constexpr int mo = 1e9 + 7;
ll fac[NUM_ + 1], factr[NUM_ + 1], inv[NUM_ + 1];

void init() {
  int i;
  inv[1] = fac[0] = factr[0] = 1;
  for (int i = 2; i <= NUM_; ++i)
    inv[i] = inv[mo % i] * (mo - mo / i) % mo;
  for (int i = 1; i <= NUM_; ++i)
    fac[i] = fac[i - 1] * i % mo, factr[i] = factr[i - 1] * inv[i] % mo;
}
ll com(ll N_, ll C_) {
  if (C_ < 0 || C_ > N_)
    return 0;
  return factr[C_] * fac[N_] % mo * factr[N_ - C_] % mo;
}
int yoko(int N, int B) {
  return (((N + 1) * com(N, B)) % mo * inv[B + 1]) % mo;
  ;
}
void solve() {
  din(H1, W1, H2, W2);
  init();
  auto calc = [&](int H, int W) {
    int res = 0;
    rep(h, H + 1) {
      if (h)
        res += yoko(W + h, h) - yoko(h - 1, h);
      else {
        res += yoko(W + h, h);
      }
      res %= mo;
    }
    return res;
  };
  int res = calc(H2, W2);
  deb(res);
  res -= calc(H1 - 1, W2);
  res %= mo;
  res += mo;
  res %= mo;
  deb(res);
  res -= calc(H2, W1 - 1);
  res %= mo;
  res += mo;
  res %= mo;
  deb(res);
  res += calc(H1 - 1, W1 - 1);
  res %= mo;
  out(res);
}

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) {
    //        cout << "完璧 : solveを書き直そう" << endl;
    //        cout << "     : そして、solve()を呼び出すのだ" << endl;
    //        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
  }
  if (sz(message)) {
    cerr << "****************************" << endl;
    cerr << message << endl;
    cerr << "****************************" << endl;
  }
#endif
  return 0;
};