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

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


#ifndef _DEBUG
#ifndef UNTITLED15_TEMPLATE_H
#define UNTITLED15_TEMPLATE_H

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

#ifndef fast_sort
#define fast_sort sort
#endif


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

using namespace std;
using namespace std::chrono;

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


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



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

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

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

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

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


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

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

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

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

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

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

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

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

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

//my_nulloptをあらゆる操作の単位元的な物として扱う
//vectorの参照外時に返したり、右辺値として渡されたときに何もしないなど
struct my_nullopt_t {} my_nullopt;
#define nullopt_t my_nullopt_t
#define nullopt my_nullopt
/*@formatter:off*/
//値が無いときは、setを使わない限り代入できない
//=を使っても無視される
template<class T> struct my_optional {
private:
    bool is_null;
    T v;
public:
    typedef T value_type ;
    my_optional() : is_null(true) {}
    my_optional(const nullopt_t&) : is_null(true) {}
    my_optional(const T& v) : v(v), is_null(false) {}
    bool has_value() const { return !is_null; }
    T &value() { static string mes = "optional has no value";assert2(!is_null, mes);return v;}
    const T &value() const { static string mes = "optional has no value";assert2(!is_null, mes);return v;}
    void set(const T &nv) {is_null = false;v = nv;}
    template<class U> void operator=(const U &v) {
        set(v);//null状態でも代入出来るようにした
//        if (has_value())value() = v; else return;
    }
    template<class U> void operator=(const my_optional<U> &v) {
        if (/*has_value() && */v.has_value())(*this) = v; else return;
    }
    /*@formatter:off*/
    void reset() { is_null = true; }
    void operator=(const nullopt_t &) { reset(); }
    template<require_t(!is_same2(T, bool))>
    explicit operator bool(){return !is_null;}
    //nullの時はエラー
    operator T&(){return value();}
    operator const T&()const {return value();}
    my_optional<T> operator++() {        if (this->has_value()) {            this->value()++;            return *this;        } else {            return *this;        }    }
    my_optional<T> operator++(signed) {        if (this->has_value()) {            auto tem = *this;            this->value()++;            return tem;        } else {            return *this;        }    }
    my_optional<T> operator--() {        if (this->has_value()) {            this->value()--;            return *this;        } else {            return *this;        }    }
    my_optional<T> operator--(signed) {        if (this->has_value()) {            auto tem = *this;            this->value()--;            return tem;        } else {            return *this;        }    }
};
template<class T>istream &operator>>(istream &iss, my_optional<T>& v) {    T val;    iss>>val;    v.set(val);    return iss;}


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

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

//vectorに対しての処理は.vを呼ぶ
template<class T> class ovv{
    optional<T> emp;
public:
    vector<vector<optional<T>> > v ;
    ovv(int i=0, int j=0, T val = 0) : v(i, vector<optional<T>>(j, val) ){}
    ovv(const vector<vector<T> >& S){
        v.resize(S.size());
        for (int h = 0; h < sz(S); h++){
            v[h].resize(sz(S[h]));
            for (int w = 0; w < sz(S[h]); w++){
                v[h][w].set(S[h][w]);
            }
        }
    }
    optional<T> &operator()(int i, int j) {        if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {            emp.reset();return emp;        }        else { return v[i][j]; }    }
    //再帰ver 遅いと思う
//    optional<T>& gets(optional<T>& v){return v;}
//    template<class V, class H, class... U> optional<T>& gets(V& v, H i, U... tail){        if constexpr(is_same2(H, nullopt_t))return operator()(-1,-1);        else if constexpr(is_same2(H, optional<int>)){            if(i.has_value())return gets(v[(int)i], tail...);            else return operator()(-1,-1);        }else if constexpr(is_integral<H>::value){            return gets(v[(int)i], tail...);        }else{            assert(0);            return emp;        }    }
#if __cplusplus >= 201703L
    //if constexprバージョン 上が遅かったらこれで
    template<class U, class V> optional<T> &operator()(const U &i, const V &j) {        /*駄目な場合を除外*/        if constexpr(is_same2(U, nullopt_t) || is_same2(U, nullopt_t)) {            return operator()(-1, -1);            /* o, o*/        } else if constexpr(is_same2(U, optional<int>) && is_same2(V, optional<int>)) {            return operator()(i.has_value() ? (int) i : -1, j.has_value() ? (int) j : -1);            /* o, x*/        } else if constexpr(is_same2(U, optional<int>)) {            return operator()(i.has_value() ? (int) i : -1, (int) j);            /* x, o*/        } else if constexpr(is_same2(V, optional<int>)) {            return operator()((int) i, j.has_value() ? (int) j : -1);            /* x, x*/        } else {            return operator()((int) i, (int) j);        }    }
#endif
    operator const vector<vector<optional<T>> >&(){
        return v;
    }
};
template<class T>istream &operator>>(istream &iss, ovv<T> &a) {    for (int h  = 0; h  < sz(a); h ++){        for (int w  = 0; w  < sz(a[h]); w ++){            iss>>a.v[h][w ];        }    }    return iss;}
template<class T>ostream &operator<<(ostream &os,  ovv<T>& a) {
    os << a.v;
    return os;
}
template<class T>string deb_tos(const ovv<T>& v){
    return deb_tos(v.v);
}
template<class T> struct ov3{
    optional<T> emp;
    vector<vector<vector<optional<T>>> > v ;
    ov3(int i, int j, int k, T val = 0) : v(i, vector<vector<optional<T>>>(j, vector<optional<T>>(k, val) ) ){}
    optional<T> &operator()(int i, int j, int k) {        if (i < 0 || j < 0 || sz(v) <= i || sz(v[i]) <= j) {            if(k < 0 || sz(v[i][j]) <= k){                emp.reset();                return emp;            }        }        return v[i][j][k];    }
private:
#if __cplusplus >= 201703L
    //再帰ver 遅いと思う
    template<class V, class H> optional<T> &gets(V &nowv, H i) {        if constexpr(is_same2(H, nullopt_t)) {            emp.reset(); return emp;        } else if constexpr(is_same2(H, optional<int>)) {            if (i.has_value()) {                return nowv[(int) i];            } else {                emp.reset();return emp;            }        } else if constexpr(is_integral<H>::value) {            return nowv[(int) i];        } else {            static string mes = "ov3 error not index";assert2(0, mes);            emp.reset();return emp;        }    }
    //todo const &消した
    template<class V, class H, class... U> optional<T> &gets(V &nowv, H i, U... tail) {        if constexpr(is_same2(H, nullopt_t)) {            emp.reset();return emp;        } else if constexpr(is_same2(H, optional<int>)) {            if (i.has_value()) {                return gets(nowv[(int) i], tail...);            } else {                emp.reset();return emp;            }        } else if constexpr(is_integral<H>::value) {            return gets(nowv[(int) i], tail...);        } else {            static string mes = "ov3 error not index";assert2(0, mes);            emp.reset();return emp;        }    }
#endif
public:
    template<class U, class V, class W> optional<T> &operator()(U i, V j, W k) {        return gets(v, i, j, k);    }
/*@formatter:off*/
};
template<class T>string deb_tos(const ov3<T>& v){
    return deb_tos(v.v);
}
//nullopt_t
//優先順位
//null, [opt, tem]

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

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

//演算子==
//optional<int>(10) == 10

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

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

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

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

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

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

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

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

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

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


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

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

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

/*@formatter:off*/

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

const double eps = 1e-9;

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


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

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

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

#define maxq 1
#define minq -1

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

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

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

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

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

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

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

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

/*@formatter:off*/
string to_string(char c) {    string ret = "";    ret += c;    return ret;}
template<class T> class pq_min_max {    vector<T> d;    void make_heap() {        for (int i = d.size(); i--;) {            if (i & 1 && d[i - 1] < d[i]) swap(d[i - 1], d[i]);            int k = down(i);            up(k, i);        }    }    inline int parent(int k) const { return ((k >> 1) - 1) & ~1; }    int down(int k) {        int n = d.size();        if (k & 1) { /* min heap*/            while (2 * k + 1 < n) {                int c = 2 * k + 3;                if (n <= c || d[c - 2] < d[c]) c -= 2;                if (c < n && d[c] < d[k]) {                    swap(d[k], d[c]);                    k = c;                }                else break;            }        }        else { /* max heap*/            while (2 * k + 2 < n) {                int c = 2 * k + 4;                if (n <= c || d[c] < d[c - 2]) c -= 2;                if (c < n && d[k] < d[c]) {                    swap(d[k], d[c]);                    k = c;                }                else break;            }        }        return k;    }    int up(int k, int root = 1) {        if ((k | 1) < (int) d.size() && d[k & ~1] < d[k | 1]) {            swap(d[k & ~1], d[k | 1]);            k ^= 1;        }        int p;        while (root < k && d[p = parent(k)] < d[k]) { /*max heap*/            swap(d[p], d[k]);            k = p;        }        while (root < k && d[k] < d[p = parent(k) | 1]) { /* min heap*/            swap(d[p], d[k]);            k = p;        }        return k;    }public:    pq_min_max() {}    pq_min_max(const vector<T> &d_) : d(d_) { make_heap(); }    template<class Iter> pq_min_max(Iter first, Iter last) : d(first, last) { make_heap(); }    void operator+=(const T &x) {        int k = d.size();        d.push_back(x);        up(k);    }    void pop_min() {        if (d.size() < 3u) { d.pop_back(); }        else {            swap(d[1], d.back());            d.pop_back();            int k = down(1);            up(k);        }    }    void pop_max() {        if (d.size() < 2u) { d.pop_back(); }        else {            swap(d[0], d.back());            d.pop_back();            int k = down(0);            up(k);        }    }    const T &get_min() const { return d.size() < 2u ? d[0] : d[1]; }    const T &get_max() const { return d[0]; }    int size() const { return d.size(); }    bool empty() const { return d.empty(); }};
//小さいほうからM個取得するpq
template<class T> struct helper_pq_size {    pq_min_max<T> q;    T su = 0;    int max_size = 0;    helper_pq_size() {}    helper_pq_size(int max_size) : max_size(max_size) {}    void clear() {        q = pq_min_max<T>();        su = 0;    }    void operator+=(T v) {        su += v;        q += (v);        if (sz(q) > max_size) {            su -= q.get_max();            q.pop_max();        }    }    T sum() { return su; }    T top() { return q.get_min(); }    void pop() {        su -= q.get_min();        q.pop_min();    }    T poll() {        T ret = q.get_min();        su -= ret;        q.pop_min();        return ret;    }    ll size() { return q.size(); }};
//大きいほうからM個取得するpq
template<class T> struct helper_pqg_size {    pq_min_max<T> q;    T su = 0;    int max_size = 0;    helper_pqg_size() {}    helper_pqg_size(int max_size) : max_size(max_size) {}    void clear() {        q = pq_min_max<T>();        su = 0;    }    void operator+=(T v) {        su += v;        q += (v);        if (sz(q) > max_size) {            su -= q.get_min();            q.pop_min();        }    }    T sum() { return su; }    T top() { return q.get_max(); }    void pop() {        su -= q.get_max();        q.pop_max();    }    T poll() {        T ret = q.get_max();        su -= ret;        q.pop_min();        return ret;    }    ll size() { return q.size(); }};;

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

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

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

#ifdef use_for

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

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

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

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

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

//#define use_for

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

#if __cplusplus >= 201703L
template<size_t M_SZ, bool indexed, class Iterator, class T, class U=T, class V=T, class W=T>
class ite_vec_merge : public Iterator {    std::size_t i = 0;    vector<T> &A;    vector<U> &B;    vector<V> &C;    vector<W> &D;public :    ite_vec_merge(Iterator ita, vector<T> &A) : Iterator(ita), A(A), B(A), C(A), D(A) {}    ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B) : Iterator(ita), A(A), B(B), C(A), D(A) {}    ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C) : Iterator(ita), A(A), B(B), C(C), D(A) {}    ite_vec_merge(Iterator ita, vector<T> &A, vector<U> &B, vector<V> &C, vector<W> &D) : Iterator(ita), A(A), B(B), C(C), D(D) {}    auto &operator++() {        ++i;        this->Iterator::operator++();        return *this;    }    auto operator*() const noexcept {        if constexpr(!indexed && M_SZ == 1) { return tuple<T &>(A[i]); }        else if constexpr(!indexed && M_SZ == 2) { return tuple<T &, U &>(A[i], B[i]); }        else if constexpr(!indexed && M_SZ == 3) { return tuple<T &, U &, V &>(A[i], B[i], C[i]); }        else if constexpr(!indexed && M_SZ == 4) { return tuple<T &, U &, V &, W &>(A[i], B[i], C[i], D[i]); }        else if constexpr(indexed && M_SZ == 1) { return tuple<int, T &>(i, A[i]); }        else if constexpr(indexed && M_SZ == 2) { return tuple<int, T &, U &>(i, A[i], B[i]); }        else if constexpr(indexed && M_SZ == 3) { return tuple<int, T &, U &, V &>(i, A[i], B[i], C[i]); }        else if constexpr(indexed && M_SZ == 4) { return tuple<int, T &, U &, V &, W &>(i, A[i], B[i], C[i], D[i]); }        else {            assert(0);            return tuple<int>(i);        }    }};
template<size_t M_SZ, bool indexed, class T, class U=T, class V=T, class W=T>
class vec_merge {    vector<T> &a;    vector<U> &b;    vector<V> &c;    vector<W> &d;public :    vec_merge(vector<T> &a) : a(a), b(a), c(a), d(a) {}    vec_merge(vector<T> &a, vector<U> &b) : a(a), b(b), c(a), d(a) {}    vec_merge(vector<T> &a, vector<U> &b, vector<V> &c) : a(a), b(b), c(c), d(a) {}    vec_merge(vector<T> &a, vector<U> &b, vector<V> &c, vector<W> &d) : a(a), b(b), c(c), d(d) {}    auto begin() const {        if constexpr(M_SZ == 1) { return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{std::begin(a), a}; }        else if constexpr(M_SZ == 2) { return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{std::begin(a), a, b}; }        else if constexpr(M_SZ == 3) { return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{std::begin(a), a, b, c}; }        else if constexpr(M_SZ == 4) { return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{std::begin(a), a, b, c, d}; }        else {            assert(0);            return ite_vec_merge<M_SZ, indexed, decltype(std::begin(a)), T, U, V, W>{std::begin(a), a};        }    }    auto end() const {        if constexpr(M_SZ == 1) { return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{std::end(a), a}; }        else if constexpr(M_SZ == 2) { return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{std::end(a), a, b}; }        else if constexpr(M_SZ == 3) { return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{std::end(a), a, b, c}; }        else if constexpr(M_SZ == 4) { return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{std::end(a), a, b, c, d}; }        else {            assert(0);            return ite_vec_merge<M_SZ, indexed, decltype(std::end(a)), T, U, V, W>{std::end(a), a};        }    }};
#endif
#define fora_2(a, A) for(auto&& a : A)

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

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

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

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

const double PI = 3.1415926535897932384626433832795029L;


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#define each for_each
#define eached for_eached


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


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

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


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

#endif

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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


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

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

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

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

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


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

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

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

#endif


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

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

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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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


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

//#define use_rui //_imo _ruic _ruiv
#ifdef use_rui
//kと同じものの数
template<class T, class U> vi imo(const vector<T> &a, U k) {    vi equ(sz(a));    rep(i, sz(a)){        equ[i] = a[i]==k;    }    return imo(equ);}
template<class T> vector<T> imox(const vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}
//漸化的に最小を持つ
template<class T> vector<T> imi(const vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}
template<class T> vector<T> ima(const vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chma(ret[i + 1], ret[i]);    return ret;}
template<class T> vector<T> rimi(const vector<T> &v) {    vector<T> ret = v;    rer(i, sz(ret) - 1, 1)chmi(ret[i - 1], ret[i]);    return ret;}
template<class T> vector<T> rima(const vector<T> &v) {    vector<T> ret = v;    rer(i, sz(ret) - 1, 1)chma(ret[i - 1], ret[i]);    return ret;}
template<class T> struct ruimax {
    template<typename Monoid> struct SegmentTree {        /*pairで処理*/        int sz;        vector<Monoid> seg;        const Monoid M1 = mp(MIN<T>(), -1);        Monoid f(Monoid a, Monoid b) { return max(a, b); }        void build(vector<T> &a) {            int n = sz(a);            sz = 1;            while (sz < n) sz <<= 1;            seg.assign(2 * sz, M1);            rep(i, n) { seg[i + sz] = mp(a[i], i); }            for (int k = sz - 1; k > 0; k--) { seg[k] = f(seg[k << 1], seg[(k << 1) | 1]); }        }        Monoid query(int a, int b) {            Monoid L = M1, R = M1;            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {                if (a & 1) L = f(L, seg[a++]);                if (b & 1) R = f(seg[--b], R);            }            return f(L, R);        }        Monoid operator[](const int &k) const { return seg[k + sz]; }    };
private:
    vector<T> ve;
    SegmentTree<pair<T, int>> seg;
    vector<T> rv;
    vector<int> ri;
    bool build = false;
public:
    int n;
    ruimax(vector<T> &a) : ve(a), n(sz(a)) {        int index = -1;        T ma = MIN<T>();        rv.resize(n + 1);        ri.resize(n + 1);        rv[0] = -INF<T>();        ri[0] = -1;        rep(i, n) {            if (chma(ma, a[i])) { index = i; }            rv[i + 1] = ma;            ri[i + 1] = index;        }    }
    T operator()(int l, int r) {        if (!(l <= r && 0 <= l && r <= n)) {            deb(l, r, n);            assert(0);        }        if (l == 0) { return rv[r]; }        else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).first;        }    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    T getv(int l, int r) { return operator()(l, r); }
    T getv(int r = inf) { return getv(0, min(r, n)); };
    int geti(int l, int r) {        assert(l <= r && 0 <= l && r <= n);        if (l == 0) { return ri[r]; }        else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).second;        }    }
    int geti(int r = inf) { return geti(0, min(r, n)); };
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};
template<class T> struct ruimin {
    template<typename Monoid> struct SegmentTree {  /*pairで処理*/         int sz;vector<Monoid> seg;        const Monoid M1 = mp(MAX<T>(), -1);        Monoid f(Monoid a, Monoid b) { return min(a, b); }        void build(vector<T> &a) {            int n = sz(a);            sz = 1;            while (sz < n) sz <<= 1;            seg.assign(2 * sz, M1);            rep(i, n) { seg[i + sz] = mp(a[i], i); }            for (int k = sz - 1; k > 0; k--) { seg[k] = f(seg[k << 1], seg[(k << 1) | 1]); }        }        Monoid query(int a, int b) {            Monoid L = M1, R = M1;            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {                if (a & 1) L = f(L, seg[a++]);                if (b & 1) R = f(seg[--b], R);            }            return f(L, R);        }        Monoid operator[](const int &k) const { return seg[k + sz]; }    };
private:
    vector<T> ve;
    SegmentTree<pair<T, int>> seg;
    vector<T> rv;
    vector<int> ri;
    bool build = false;
    int n;
public:
    ruimin(vector<T> &a) : ve(a), n(sz(a)) {        int index = -1;        T mi = MAX<T>();        rv.resize(n + 1);        ri.resize(n + 1);        rv[0] = INF<T>();        ri[0] = -1;        rep(i, n) {            if (chmi(mi, a[i])) { index = i; }            rv[i + 1] = mi;            ri[i + 1] = index;        }    }
    T operator()(int l, int r) {        assert(l <= r && 0 <= l && r <= n);        if (l == 0) { return rv[r]; }        else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).first;        }    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    T getv(int l, int r) { return operator()(l, r); }
    T getv(int r = inf) { return getv(0, min(r, n)); };
    int geti(int l, int r) {        {            assert(l <= r && 0 <= l && r <= n);            if (l == 0) { return ri[r]; }            else {                if (!build)seg.build(ve), build = true;                return seg.query(l, r).second;            }        }        assert(l <= r && 0 <= l && r <= n);        if (l == 0) { return ri[r]; }        else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).second;        }    }
    int geti(int r = inf) { return geti(0, min(r, n)); };
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};/*@formatter:off*/

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

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

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


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

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

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

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

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

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

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

//forsubをスニペットして使う
//Mの部分集合(0,M含む)を見る 3^sz(S)個ある
#define forsub_all(m, M) for (int m = M; m != -1; m = m == 0 ? -1 : (m - 1) & M)
//BASE進数
template<size_t BASE> class base_num {
    int v;
public:
    base_num(int v = 0) : v(v) {};
    int operator[](int i) { return bget(v, i, BASE); }
    void operator++() { v++; }
    void operator++(signed) { v++; }
    operator int() { return v; }
};
#define base3(mas, lim, BASE) for (base_num<BASE> mas; mas < lim; mas++)
#define base2(mas, lim) base3(mas, lim, 2)
#define for_base(...) over3(__VA_ARGS__,base3,base2,base1)(__VA_ARGS__)


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


/*@formatter:off*/

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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


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

#endif /*UNTITLED15_TEMPLATE_H*/


#endif

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

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

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

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

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

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


void solve() {
    in(N, M);
    mint res = 0;
    rep(same, 0, N + 1) {
        mint pat = 1;
        //場所
        pat *= com(N, same);
        //一致で使う色
        pat *= com(M, same);
        //一致色の並び
        pat *= fac[same];

        //残り、上使う色
        pat *= com(M - same, N - same);
        //残り、下使う色
        pat *= com(M - same, N - same);

        //残り、上の色の並び
        pat *= fac[N - same];
        //のこり、下の色の並び
        pat *= fac[N - same];

        if (same % 2 == 0)res += pat;
        else res -= pat;
    }
    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) {
        //solveを書き直す
        //solveを呼び出す
    }
    exit(0);
#endif
    return 0;
};


Measured memory_bytes: 36304