/*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>
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <boost/multiprecision/cpp_int.hpp>
using cint = boost::multiprecision::cpp_int;
using bdou = boost::multiprecision::cpp_dec_float_100;

#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
//#include "template.h"
#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>
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

using namespace std;
using namespace std::chrono;


/*@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>;
#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();
    }
};
#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
//任意のマクロサポート用 使う度に初期化する
int index_, v1_, v2_, v3_;

/*@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;}

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
#if __cplusplus >= 201703L
//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);});}
/*@formatter:off*/
//存在しない場合
//1次元 Nを返す
//多次元-1を返す
template<typename T, typename F> ll find_if2(const vector<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> ll rfind_if2(const vector<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);    }}
#else
template<typename T, typename F> bool all_of2(const T &v, F f) { return f(v); }
template<typename T, typename F> bool all_of2(const vector<T> &v, F f) {    rep(i, sz(v)) { if (!all_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool any_of2(const T &v, F f) { return f(v); }
template<typename T, typename F> bool any_of2(const vector<T> &v, F f) {    rep(i, sz(v)) { if (any_of2(v[i], f))return true; }    return false;}
template<typename T, typename F> bool none_of2(const T &v, F f) { return f(v); }
template<typename T, typename F> bool none_of2(const vector<T> &v, F f) {    rep(i, sz(v)) { if (none_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool find_if2(const T &v, F f) { return f(v); }
template<typename T, typename F> ll find_if2(const vector<T> &v, F f) {    rep(i, sz(v)) { if (find_if2(v[i], f))return i; }    return sz(v);}
template<typename T, typename F> bool rfind_if2(const T &v, F f) { return f(v); }
template<typename T, typename F> ll rfind_if2(const vector<T> &v, F f) {    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }    return -1;}
template<class T> bool contains(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) { return f(v); }
template<typename T, typename F> ll count_if2(const vector<T> &vec_, F f) {    ll ret = 0;    fora(v, vec_) { ret += count_if2(v, f); }    return ret;}

template<typename T, typename F> void for_each2(T &a, F f) {
    f(a);
}
template<typename T, typename F> void for_each2(vector<T>  &a, F f) {
    for (auto &&v_ : a)for_each2(v_, f);
}
#endif
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

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

template<class T> bool 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
template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;    rep(i, sz(a))res.eb(a[i], b[i]);    return res;}
template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {    a.resize(sz(p)), b.resize(sz(p));    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;}
//vector to tuple
template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {    assert(sz(a) == sz(b) && sz(b) == sz(c));    vector<tuple<A, B, C>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i]);    return res;}
template<class A, class B, class C, class D> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));    vector<tuple<A, B, C, D>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i], d[i]);    return res;}

template<class A, class B, class C> auto ttov(const vector<tuple<A, B, C>> &v) {    tuple<vector<A>, vector<B>, vector<C>> res;    for (int i = 0; i < v.size(); i++) {        get<0>(res).push_back(get<0>(v[i]));        get<1>(res).push_back(get<1>(v[i]));        get<2>(res).push_back(get<2>(v[i]));    }    return res;}
template<class A, class B, class C, class D> auto ttov(const vector<tuple<A, B, C, D>> &v) {    tuple<vector<A>, vector<B>, vector<C>, vector<D>> res;    for (int i = 0; i < v.size(); i++) {        get<0>(res).push_back(get<0>(v[i]));        get<1>(res).push_back(get<1>(v[i]));        get<2>(res).push_back(get<2>(v[i]));        get<3>(res).push_back(get<3>(v[i]));    }    return res;}
/*@formatter:off*/
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>());};
template<class A, class B> void sortp(vector<A> &a, vector<B> &b) {    auto c = vtop(a, b);    sort(c);    rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;}
template<class A, class B> void rsortp(vector<A> &a, vector<B> &b) {    auto c = vtop(a, b);    rsort(c);    rep(i, sz(a))a[i] = c[i].first, b[i] = c[i].second;}
template<class A, class B, class C> void sortt(vector<A> &a, vector<B> &b, vector<C> &c) {    auto d = vtot(a, b, c);    sort(d);    tie(a, b, c) = ttov(d);}
template<class A, class B, class C> void rsortt(vector<A> &a, vector<B> &b, vector<C> &c) {    auto d = vtot(a, b, c);    rsort(d);    tie(a, b, c) = ttov(d);}
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;}
//sortindex 元のvectorはソートしない
template<class T> vi sorti(vector<T> &a) {    auto b = a;    vi ind = iota(0, sz(a));    sortp(b, ind);    return ind;}
/*@formatter:off*/
//#define use_sort
#ifdef use_sort
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {    fisiti,    fisitd,    fisdti,    fisdtd,    fdsiti,    fdsitd,    fdsdti,    fdsdtd,    fitisi,    fitisd,    fitdsi,    fitdsd,    fdtisi,    fdtisd,    fdtdsi,    fdtdsd,    sifiti,    sifitd,    sifdti,    sifdtd,    sdfiti,    sdfitd,    sdfdti,    sdfdtd,    sitifi,    sitifd,    sitdfi,    sitdfd,    sdtifi,    sdtifd,    sdtdfi,    sdfdfd,    tifisi,    tifisd,    tifdsi,    tifdsd,    tdfisi,    tdfisd,    tdfdsi,    tdfdsd,    tisifi,    tisifd,    tisdfi,    tisdfd,    tdsifi,    tdsifd,    tdsdfi,    tdsdfd};
template<class U> void sort(vector<U> &a, pcomparator type) {    if (type == fisi) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) < get<1>(r); });    else if (type == fisd) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) > get<1>(r); }); else if (type == fdsi) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) < get<1>(r); }); else if (type == fdsd) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) > get<1>(r); }); else if (type == sifi) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) < get<0>(r); }); else if (type == sifd) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) > get<0>(r); }); else if (type == sdfi) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) < get<0>(r); }); else if (type == sdfd) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) > get<0>(r); });};
template<class U> void sort(vector<U> &a, tcomparator type) {    if (type == 0) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) < get<2>(r); });    else if (type == 1) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) > get<2>(r); });    else if (type == 2) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) < get<2>(r); });    else if (type == 3) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) > get<2>(r); });    else if (type == 4) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) < get<2>(r); });    else if (type == 5) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) > get<2>(r); });    else if (type == 6) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) < get<2>(r); });    else if (type == 7) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) > get<2>(r); });    else if (type == 8) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) < get<1>(r); });    else if (type == 9) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) > get<1>(r); });    else if (type == 10) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) < get<1>(r); });    else if (type == 11) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) > get<1>(r); });    else if (type == 12) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) < get<1>(r); });    else if (type == 13) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) > get<1>(r); });    else if (type == 14) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) < get<1>(r); });    else if (type == 15) sort(ALL(a), [&](U l, U r) { return get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) > get<1>(r); });    else if (type == 16) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) < get<2>(r); });    else if (type == 17) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) > get<2>(r); });    else if (type == 18) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) < get<2>(r); });    else if (type == 19) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) > get<2>(r); });    else if (type == 20) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) < get<2>(r); });    else if (type == 21) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<2>(l) > get<2>(r); });    else if (type == 22) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) < get<2>(r); });    else if (type == 23) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<2>(l) > get<2>(r); });    else if (type == 24) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) < get<0>(r); });    else if (type == 25) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) > get<0>(r); });    else if (type == 26) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) < get<0>(r); });    else if (type == 27) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) > get<0>(r); });    else if (type == 28) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) < get<0>(r); });    else if (type == 29) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) > get<0>(r); });    else if (type == 30) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) < get<0>(r); });    else if (type == 31) sort(ALL(a), [&](U l, U r) { return get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) > get<0>(r); });    else if (type == 32) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) < get<1>(r); });    else if (type == 33) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) > get<1>(r); });    else if (type == 34) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) < get<1>(r); });    else if (type == 35) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) > get<1>(r); });    else if (type == 36) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) < get<1>(r); });    else if (type == 37) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) < get<0>(r) : get<1>(l) > get<1>(r); });    else if (type == 38) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) < get<1>(r); });    else if (type == 39) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<0>(l) != get<0>(r) ? get<0>(l) > get<0>(r) : get<1>(l) > get<1>(r); });    else if (type == 40) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) < get<0>(r); });    else if (type == 41) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) > get<0>(r); });    else if (type == 42) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) < get<0>(r); });    else if (type == 43) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) < get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) > get<0>(r); }); else if (type == 44) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) < get<0>(r); }); else if (type == 45) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) < get<1>(r) : get<0>(l) > get<0>(r); }); else if (type == 46) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) < get<0>(r); }); else if (type == 47) sort(ALL(a), [&](U l, U r) { return get<2>(l) != get<2>(r) ? get<2>(l) > get<2>(r) : get<1>(l) != get<1>(r) ? get<1>(l) > get<1>(r) : get<0>(l) > get<0>(r); });}
/*@formatter:off*/
/*@formatter:off*/
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);}
//P l, P rで f(P) の形で渡す
template<class U, class F> void sort(vector<U> &a, F f) { sort(ALL(a), [&](U l, U r) { return f(l) < f(r); }); };
template<class U, class F> void rsort(vector<U> &a, F f) { sort(ALL(a), [&](U l, U r) { return f(l) > f(r); }); };
//F = T<T>
//例えばreturn p.fi + p.se;
template<class A, class B, class F> void sortp(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    sort(c, f);    rep(i, sz(a)) a[i] = c[i].fi, b[i] = c[i].se;}
template<class A, class B, class F> void rsortp(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    rsort(c, f);    rep(i, sz(a))a[i] = c[i].first, b[i] = c[i].second;}
template<class A, class B, class C, class F> void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    sort(d, f);    tie(a, b, c) = ttov(d);}
template<class A, class B, class C, class F> void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    rsort(d, f);    tie(a, b, c) = ttov(d);}
template<class A, class B, class C, class D> void sortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {    auto e = vtot(a, b, c, d);    sort(e);    tie(a, b, c, d) = ttov(e);}
template<class A, class B, class C, class D> void rsortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {auto e = vtot(a, b, c, d);rsort(e);tie(a, b, c, d) = ttov(e);}
/*@formatter:off*/
/*indexの分で型が変わるためpcomparatorが必要*/
template<class T> vi sorti(vector<T> &a, pcomparator f) {    auto b = a;    vi ind = iota(0, sz(a));    sortp(b, ind, f);    return ind;}
template<class T, class F> vi sorti(vector<T> &a, F f) {    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(a[x]) < f(a[y]); });    return ind;}
template<class T> vi rsorti(vector<T> &a) {    auto b = a;    vi ind = iota(0, sz(a));    rsortp(b, ind);    return ind;}
template<class T, class F> vi rsorti(vector<T> &a, F f) {    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(a[x]) > f(a[y]); });    return ind;}
template<class A, class B, class F> vi sortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(c[x]) < f(c[y]); });    return ind;}
template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {    vi ind = iota(0, sz(a));    auto c = a;    auto d = b;    sortt(c, d, ind, f);    return ind;}
template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fisi); };
template<class A, class B, class F> vi rsortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(c[x]) > f(c[y]); });    return ind;}
template<class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fdsd); };
template<class A, class B, class C, class F> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(d[x]) < f(d[y]); });    return ind;}
template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {    vi ind = iota(0, sz(a));    auto d = vtot(a, b, c, ind);    sort(d, f);    rep(i, sz(a))ind[i] = d[i].d;    return ind;}
template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { if (a[x] == a[y]) { if (b[x] == b[y])return c[x] < c[y]; else return b[x] < b[y]; } else { return a[x] < a[y]; }});    return ind;}
template<class A, class B, class C, class F> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { return f(d[x]) > f(d[y]); });    return ind;}
template<class A, class B, class C> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(ALL(ind), [&](ll x, ll y) { if (a[x] == a[y]) { if (b[x] == b[y])return c[x] > c[y]; else return b[x] > b[y]; } else { return a[x] > a[y]; }});    return ind;}
template<class T> void sort2(vector<vector<T >> &a) { for (ll i = 0, n = a.size(); i < n; ++i)sort(a[i]); }
template<class T> void rsort2(vector<vector<T >> &a) { for (ll i = 0, n = a.size(); i < n; ++i)rsort(a[i]); }

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

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

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

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

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

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

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

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

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


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


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

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

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

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

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


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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


/*@formatter:off*/

//機能拡張
#define dtie(a, b) int a, b; tie(a, b)
template<class T, class U> string to_string(T a, U b) {    string res = "";    res += a;    res += b;    return res;}
template<class T, class U, class V> string to_string(T a, U b, V c) {    string res = "";    res += a;    res += b;    res += c;    return res;}
template<class T, class U, class V, class W> string to_string(T a, U b, V c, W d) {    string res = "";    res += a;    res += b;    res += c;    res += d;    return res;}
template<class T, class U, class V, class W, class X> string to_string(T a, U b, V c, W d, X e) {    string res = "";    res += a;    res += b;    res += c;    res += d;    res += e;    return res;}
//todo stringもセットで
//find_ifはsubrを使う
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); }
template<class T> vector<T> subn(const vector<T> &A, int l, int len) { return sub(A, l, l + len); }
string sub(string &A, int l, int r) {    assert(0 <= l && l <= r && r <= sz(A));    return A.substr(l, r - l);}
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, 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, 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, U v) { a.push(v); }template<class T, class U> void operator+=(deque<T> &a, U v) { a.push_back(v); }template<class T> priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, vector<T> &v) {    fora(d, v){a.push(d);}    return a;}template<class T, class U> priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, U v) {    a.push(v);    return a;}template<class T, class U> priority_queue<T> &operator+=(priority_queue<T> &a, U v) {    a.push(v);    return a;}template<class T> set<T> &operator+=(set<T> &a, vector<T> v) {    fora(d, v){a.insert(d);}    return a;}template<class T, class U> auto operator+=(set<T> &a, U v) { return a.insert(v); }template<class T, class U> auto operator-=(set<T> &a, U v) { return a.erase(v); }template<class T, class U> auto operator+=(mset<T> &a, U v) { return a.insert(v); }template<class T, class U> set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {    a.insert(v);    return a;}template<class T, class U> vector<T> &operator+=(vector<T> &a, U v) {    a.push_back(v);    return a;}template<class T, class U> vector<T> operator+(U v,const vector<T> &a) {    vector<T> ret = a;    ret.insert(ret.begin(), v);    return ret;}template<class T> vector<T> operator+(const vector<T>& a, const vector<T>& b) {    vector<T> ret;    ret = a;    fora(v, b){ret += v; }   return ret;}template<class T> vector<T> &operator+=(vector<T> &a,const vector<T> &b) {    rep(i, sz(b)) {/*こうしないとa+=aで両辺が増え続けてバグる*/        a.push_back(b[i]); }    return a;}template<class T, class U> map<T, U> &operator+=(map<T, U> &a, map<T, U> &b) {    for(auto&& bv : b) { a[bv.first] += bv.second; }    return a;}template<class T, class U> vector<T> operator+(const vector<T> &a, const U& v) {    vector<T> ret = a;    ret += v;    return ret;}template<class T, class U> auto operator+=(uset<T> &a, U v) { return a.insert(v); }
template<class T> vector<T> operator%(vector<T>& a, int v){    vi ret(sz(a));    rep(i,sz(a)){        ret[i] = a[i] % v;    }    return ret;}
template<class T> vector<T> operator%=(vector<T>& a, int v){    rep(i,sz(a)){        a[i] %= v;    }    return a;}
vi operator&(vi& a, vi& b){    assert(sz(a)==sz(b));    vi ret(sz(a));    rep(i,sz(a)){        ret[i] = min(a[i],b[i]);    }    return ret;}
template<class T> void operator+=(mset<T> &a, vector<T>& v) { for(auto&& u : v)a.insert(u); }
template<class T> void operator+=(set<T> &a, vector<T>& v) { for(auto&& u : v)a.insert(u); }
template<class T> void operator+=(vector<T> &a, set<T>& v) { for(auto&& u : v)a.emplace_back(u); }
template<class T> void operator+=(vector<T> &a, mset<T>& v) { for(auto&& u : v)a.emplace_back(u); }
template<class T> vector<T> &operator-=(vector<T> &a, vector <T> &b) {    if (sz(a) != sz(b)) {        debugline("vector<T> operator-=");        deb(a);        deb(b);        exit(0);    }    rep(i, sz(a))a[i] -= b[i];    return a;}
template<class T> vector<T> operator-(vector<T> &a, vector<T> &b) {    if (sz(a) != sz(b)) {        debugline("vector<T> operator-");        deb(a);        deb(b);        ole();    }    vector<T> res(sz(a));    rep(i, sz(a))res[i] = a[i] - b[i];    return res;}
//template<class T, class U> void operator*=(vector<T> &a, U b) {    vector<T> ta = a;    rep(b-1){        a+=ta;    }}
template<typename T> void erase(T &v, unsigned ll i) { v.erase(v.begin() + i); }
template<typename T> void erase(T &v, unsigned ll s, unsigned ll e) { v.erase(v.begin() + s, v.begin() + e); }
template<class T, class... I> auto erased(T& v, I... arg){    auto ret = v;    erase(ret, arg...);    return ret;}
template<typename T> void pop_front(vector<T> &v) { erase(v, 0); }
template<typename T> void entry(vector<T> &v, unsigned ll s, unsigned ll e) { erase(v, e, sz(v));erase(v,0,s);}

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

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

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


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

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

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

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

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

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

/*@formatter:off*/
// ss << rhs で型によらず追加できる
// ss + rhs が上を返す
struct s_stream : public basic_string<char, char_traits<char>, allocator<char>> {
    using STR = basic_string<char, char_traits<char>, allocator<char>>;
    s_stream() : STR() {}
    template<class T>
    s_stream(const T &v) : STR() {operator<<(v);}
    template<class T>s_stream &operator<<(const T &rhs) {        stringstream ss;        ss << rhs;        (*this) += ss.str();        return *this;    }
    template<class T>    s_stream operator+(const T &rhs) {        auto res = *this;        res << rhs;        return res;    }
    string str() {        return *this;    }
};/*@formatter:on*/
#if __cplusplus >= 201703L
//template<class T> auto sum(int a, T  v...) {return (v + ... + 0);}
#endif
#define VEC vector
#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;
namespace {/*@formatter:off*/
#undef getid
#undef getid_1
#undef getid_2

//#define type_id_graph
#ifdef type_id_graph
//#define forg_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(), f, t, c,ty,id; gi < forg_flim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c,ty=ve[gi].ty, id = ve[gi].id, true); ++gi)
#define forg_f_init(ve) int f = ve[gi].f, t = ve[gi].t, ty = ve[gi].ty, id = ve[gi].id; auto& c = ve[gi].c
#define forg_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(); gi < forg_flim; ++gi)
#define fort_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(); gi < forg_flim ; ++gi)
#define fort_f_init(gi, ve) int f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, ty = ve[gi].ty, id = ve[gi].id;if(t == p)continue;
//#define fort_f(gi, ve) for (ll gi = 0, f, t, c, ty, id,wasp = (p != -1 && ve[gi].t == p) ? 1 : 0; (wasp = wasp ? 1 : (p != -1 && ve[gi].t == p) ? 1 : 0,gi + wasp < ve.size()) &&(tie(f, t, c, ty, id) = wasp ?  make_tuple(ve[gi + 1].f, ve[gi + 1].t, ve[gi + 1].c, ve[gi + 1].ty, ve[gi + 1].id) : make_tuple(ve[gi].f, ve[gi].t, ve[gi].c, ve[gi].ty, ve[gi].id), true); ++gi)
#define fore(gi, ve) for (ll gi = 0,forg_flim = ve.size(), f, t, c, ty, id; gi < forg_flim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, ty = ve[gi].ty, id = ve[gi].id, true); ++gi)

template<class T> struct edge_ {        typedef T value_type;int f, t, ty, id;    T c;    edge_(int f, int t, T c = 1, int ty = -1, int id= -1) : f(f), t(t), c(c), ty(ty), id(id) {}    bool operator<(const edge_ &b) const { return c < b.c; }    bool operator>(const edge_ &b) const { return c > b.c; }};
template<class T> ostream &operator<<(ostream &os, edge_<T> &e) {os << e.f << " " << e.t << " " << e.c<<" "<< e.ty<<" "<<e.id;return os;}
template<typename T> class graph {
public :
    typedef T value_type;
    vector<vector<edge_<T>>> g;
    vector<edge_<T>> edges;
    int n;
    explicit graph(int n) : n(n) { g.resize(n); }
    void clear() { g.clear(), edges.clear(); }
    void resize(int n) {this->n = n;g.resize(n);}
    int size() { return n; }
    vector<edge_<T> > &operator[](int i) { return g[i]; }
    virtual void add(int f, int t, T c, int ty, int id) = 0;
    virtual void set_edges() = 0;
};
template<typename T =ll> class digraph : public graph<T> {
public:
    using graph<T>::g;
    using graph<T>::n;
    using graph<T>::edges;
    explicit digraph(int n) : graph<T>(n) {}
    explicit digraph(int n, const vector<edge_<T>>& E) : graph<T>(n) {        fora(e,E){            add(e.f,e.t,e.c);        }    }
    void add(int f, int t, T c = 1, int ty = -1, int eid = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {debugline("digraph add");deb(f, t, c);ole();}
        int id = sz(edges);
        if(eid != -1)id = eid;
        g[f].emplace_back(f, t, c, ty, id);
        edges.emplace_back(f, t, c, ty, id);//edgesを使わないなら消せる
    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges() override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
template<class T=int> class undigraph : public graph<T> {
public:
    using graph<T>::g;    using graph<T>::n;    using graph<T>::edges;
    explicit undigraph(int n) : graph<T>(n) {}
    explicit undigraph(int n, const vector<edge_<T>>& E) : graph<T>(n) {        fora(e,E){            add(e.f,e.t,e.c);        }    }
    int next_id = 0;
    // f < t
    void add(int f, int t, T c = 1, int ty = -1, int eid = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("undigraph add");            deb(f, t, c);            ole();        }
        int id = eid;
        if(id == -1){
            id = next_id++;
        }
        g[f].emplace_back(f, t, c, ty, id);
        g[t].emplace_back(t, f, c, ty, id);
        edges.emplace_back(f, t, c, ty, id);//edgesを使わないなら消せる
        edges.emplace_back(t, f, c, ty, id);
    }
    void add(edge_<T> &e) {        int f = e.f, t = e.t,ty = e.ty,id=e.id;        T c = e.c;        add(f, t, c, ty, id);    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges () override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
#else //#ifdef type_id_graph
#define forg_f_init(ve) int f = ve[gi].f, t = ve[gi].t; auto& c = ve[gi].c
#define forg_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(); gi < forg_flim; ++gi)
// /*temp2*/#define forg_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(), f, t, c; gi < forg_flim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)
//#define fort_f(gi, ve) for (ll gi = 0, f, t, c,wasp = (p != -1 && ve[gi].t == p) ? 1 : 0; (wasp = wasp ? 1 : (p != -1 && ve[gi].t == p) ? 1 : 0,gi + wasp < ve.size()) &&(tie(f, t, c) = wasp ?  make_tuple(ve[gi + 1].f, ve[gi + 1].t, ve[gi + 1].c) : make_tuple(ve[gi].f, ve[gi].t, ve[gi].c), true); ++gi)
#define fort_f(gi, ve) for (ll gi = 0,forg_flim = ve.size(); gi < forg_flim ; ++gi)
#define fort_f_init(gi, ve) int f = ve[gi].f, t = ve[gi].t, c = ve[gi].c;if(t == p)continue;
#define fore(gi, ve) for (ll gi = 0,forg_flim = ve.size(), f, t, c; gi < forg_flim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)

template<class T> struct edge_ {typedef T value_type;int f, t;    T c;    edge_(int f, int t, T c = 1) : f(f), t(t), c(c) {}    bool operator<(const edge_ &b) const { return c < b.c; }    bool operator>(const edge_ &b) const { return c > b.c; }};
template<class T> ostream &operator<<(ostream &os, edge_<T> &e) {os << e.f << " " << e.t << " " << e.c;return os;}
template<typename T> class graph {
public :
    typedef T value_type;
    vector<vector<edge_<T>>> g;
    vector<edge_<T>> edges;
    int n;
    explicit graph(int n) : n(n) { g.resize(n); }
    void clear() { g.clear(), edges.clear(); }
    void resize(int n) {this->n = n;g.resize(n);}
    int size() { return n; }
    vector<edge_<T> > &operator[](int i) { return g[i]; }
    virtual void add(int f, int t, T c) = 0;
    virtual void set_edges() = 0;
};
template<typename T =ll> class digraph : public graph<T> {
public:
    using graph<T>::g;
    using graph<T>::n;
    using graph<T>::edges;
    explicit digraph(int n) : graph<T>(n) {}
    explicit digraph(int n, const vector<edge_<T>>& E) : graph<T>(n) {        fora(e,E){            add(e.f,e.t,e.c);        }    }
    void add(int f, int t, T c = 1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {debugline("digraph add");deb(f, t, c);ole();}
        g[f].emplace_back(f, t, c);
        edges.emplace_back(f, t, c);//edgesを使わないなら消せる
    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges() override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
template<class T=int> class undigraph : public graph<T> {
public:
    using graph<T>::g;    using graph<T>::n;    using graph<T>::edges;
    explicit undigraph(int n) : graph<T>(n) {}
    explicit undigraph(int n, const vector<edge_<T>>& E) : graph<T>(n) {        fora(e,E){            add(e.f,e.t,e.c);        }    }
    // f < t
    void add(int f, int t, T c = 1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("undigraph add");            deb(f, t, c);            ole();        }
        g[f].emplace_back(f, t, c);
        g[t].emplace_back(t, f, c);
        edges.emplace_back(f, t, c);//edgesを使わないなら消せる
        edges.emplace_back(t, f, c);
    }
    void add(edge_<T> &e) {        int f = e.f, t = e.t;        T c = e.c;        add(f, t, c);    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges () override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
#endif

#define dijkstra_path dis_path
template<class T,class I> vi dis_path(digraph<T> &g, vector<T> &dis, int s, int t, I init_value) {    assert(dis[t] != init_value);    auto rg = rev(g);    int now = t;    vi path;    path.push_back(now);    T cost = 0;    while (now != s) {        rep(gi, sz(rg[now])) {            int m = rg[now][gi].t;            if (dis[m] == init_value)continue;            if (dis[m] + rg[now][gi].c + cost == dis[t]) {                cost += rg[now][gi].c;                now = m;                break;            }        }        path.push_back(now);    }    rev(path);    return path;}
template<class T,class I> vi dis_path(undigraph<T> &g, vector<T> &dis, int s, int t, I init_value) {    assert(dis[t] != init_value);    int now = t;    vi path;    path.push_back(now);    T cost = 0;    while (now != s) {        rep(gi, sz(g[now])) {            int m = g[now][gi].t;            if (dis[m] == init_value)continue;            if (dis[m] + g[now][gi].c + cost == dis[t]) {                cost += g[now][gi].c;                now = m;                break;            }        }        path.push_back(now);    }    rev(path);    return path;}
template<class T,class I> vi dis_path(digraph<T> &g, vector<vector<T>> &dis, int s, int t, I init_value) { return dis_path(g, dis[s], s, t, init_value); }
template<class T,class I> vi dis_path(undigraph<T> &g, vector<vector<T>> &dis, int s, int t, I init_value) { return dis_path(g, dis[s], s, t, init_value); }
vi to_vi(const int& i){return vi{i};}
vi to_vi(const vi& v){return v;}
#define dijk dijkstra
//O(N^2)
template<class T, class U, class I> vector<T> dijkstra_mitu(graph<T> &g, const U &S_, I init_value) {    vi S = to_vi(S_);    int n = g.n;    T initValue = MAX<T>();    vector<T> dis(n, initValue);    fora(s, S) {        if (!(0 <= s && s < g.n)) {            debugline("dijkstra_mitu");            deb(s, g.n);            ole();        }        dis[s] = 0;    }    vb used(n);    while (true) {        int v = -1;        rep(u, n) { if (!used[u] && (v == -1 || dis[u] < dis[v]))v = u; }        if (v == -1)break;        if (dis[v] == initValue)break;        used[v] = true;        rep(u, sz(g[v])) {            auto e = g[v][u];            dis[e.t] = min(dis[e.t], dis[v] + e.c);        }    }    /*基本、たどり着かないなら-1*/    for (auto &&d :dis) { if (d == initValue) { d = init_value; }}    return dis;}
template<class T, class U, class I> vector<T> dijkstra_01(graph<T> &g, const U &S_, I init_value) {int N = g.n;    vi S = to_vi(S_);    T tinf = INF<T>();    vector<T> dis(N, tinf);    deque<int> q;    fora(s, S) {        dis[s] = 0;        q.push_back(s);    }    vb was(N);    while (!q.empty()) {        int f = q.front();        q.pop_front();        if (was[f])continue;        was[f] = true;        fora(e, g[f]) {            if (dis[e.t] > dis[f] + e.c) {                if (e.c) {                    dis[e.t] = dis[f] + 1;                    q.push_back(e.t);                }                else {                    dis[e.t] = dis[f];                    q.push_front(e.t);                }            }        }    }    rep(i, N)if (dis[i] == tinf)dis[i] = init_value;    return dis;}
template<typename T> struct radixheap {    vector<pair<u64, T> > v[65];    u64 size_, last;    radixheap() : size_(0), last(0) {}    bool empty() const { return size_ == 0; }    int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }    void emplace(u64 key, const T &value) {        ++size_;        v[getbit(key ^ last)].emplace_back(key, value);    }    void push(u64 key, const T &value) { emplace(key, value); }    pair<u64, T> top() {        if (v[0].empty()) {            int idx = 1;            while (v[idx].empty()) ++idx;            last = min_element(begin(v[idx]), end(v[idx]))->first;            for (auto &p : v[idx]) v[getbit(p.first ^ last)].emplace_back(p);            v[idx].clear();        }        --size_;        auto ret = v[0].back();        v[0].pop_back();        return ret;    }    void pop() { ; }    int size() { return size_; }};
//01, 密グラフ, normalを自動で判断する
template<class Q, class T, class U,class I> vector<T> private_dijkstra(graph<T> &g, U &S_, I init_value) {    vi S = to_vi(S_);    fora(s, S) {        if (!(0 <= s && s < g.n)) {            debugline("dijkstra");            deb(s, g.n);            ole();        }    }    bool cost01 = true;    rep(i, g.n) { forg_f(gi, g[i]) {forg_f_init(g[i]); cost01 &= (c == 0 || c == 1); }}    if (cost01)return dijkstra_01(g, S_, init_value); else if ((g.n + sz(g.edges)) * log2(g.n) > g.n * g.n) { return dijkstra_mitu(g, S, init_value); }    T initValue = MAX<T>();    vector<T> dis(g.n, initValue);    Q q;    fora(s, S) {        dis[s] = 0;        q.emplace(0, s);    }    while (q.size()) {        T nowc;        int i;        tie(nowc, i) = q.top();        q.pop();        if (dis[i] != nowc)continue;        for (auto &&e  : g.g[i]) {            int to = e.t;            T c = nowc + e.c;            if (dis[to] > c) {                dis[to] = c;                q.emplace(dis[to], to);            }        }    }    /*基本、たどり着かないなら-1*/    for (auto &&d :dis) if (d == initValue)d = init_value;    return dis;}
template<class T, class U,class I> vector<T> dijkstra(graph<T> &g, U S_, I init_value) { if (typeid(T) == typeid(int)) { return private_dijkstra<radixheap<int>>(g, S_, init_value); } else { return private_dijkstra<priority_queue<pair<T, int>, vector<pair<T, int >>, greater<pair<T, int >> >>(g, S_, init_value); }}
//dijkstra_cou<mint> : 数える型で書く return vp(dis,cou)
template<class COU,class T=int,class I> auto dijkstra_cou(const graph<T> &g, int s, I init_value) {    if (!(0 <= s && s < g.n)) {        debugline("dijkstra");        deb(s, g.n);        ole();    }    err("count by type COU ");    err("int or mint");    T initValue = MAX<T>();    vector<T> dis(g.n, initValue);    vector<COU> cou(g.n);    cou[s] = 1;    priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> q;    dis[s] = 0;    q.emplace(0, s);    while (q.size()) {        T nowc = q.top().fi;        int i = q.top().se;        q.pop();        if (dis[i] != nowc)continue;        for (auto &&e  : g.g[i]) {            int to = e.t;            T c = nowc + e.c;            if (dis[to] > c) {                dis[to] = c;                cou[to] = cou[e.f];                q.emplace(dis[to], to);            }            else if (dis[to] == c) { cou[to] += cou[e.f]; }        }    }    /*基本、たどり着かないなら-1*/    for (auto &&d :dis) if (d == initValue)d = init_value;    return vtop(dis, cou);}
//コストを無限に減らせる := -linf
//たどり着けない := linf
template<class T> vector<T> bell(graph<T> &g, int s) {    if (g.n >= 1e4) {        cout << "bell size too big" << endl;        exit(0);    }    vector<T> res(g.n, linf);    res[s] = 0;    vb can(g.n);    /*頂点から行けない頂点を持つ、辺を消しておく */    fix([&](auto ds, int p, int i) -> void {        if (can[i])return;        can[i] = true;            forg_f(gi, g[i]){                forg_f_init(g[i]);                if (t != p)ds(i, t);            }        })(-1, 0);        vector<edge_<T>> es;        fora(e, g.edges) { if (can[e.f])es += e; }        rep(i, g.n) {            bool upd = false;            fora(e, es) {                if (res[e.f] != linf && res[e.t] > res[e.f] + e.c) {                    upd = true;                    res[e.t] = res[e.f] + e.c;                }            }            if (!upd)break;        }        rep(i, g.n * 2) {            bool upd = 0;            fora(e, g.edges) {                if (res[e.f] != linf && res[e.t] != -linf && res[e.t] > res[e.f] + e.c) {                    upd = 1;                    res[e.t] = -linf;                }            }            if (!upd)break;        }        return res;    }
//コストを無限に増やせる := linf
//たどり着けない := -linf
template<class T> vector<T> bell_far(graph<T> &g, int s) {        if (g.n >= 1e4) {            cout << "bell_far size too big" << endl;            exit(0);        }        vector<T> res(g.n, linf);        res[s] = 0;        vb can(g.n);    /*頂点から行けない頂点を持つ、辺を消しておく*/    fix([&](auto ds, int p, int i) -> void {            if (can[i])return;            can[i] = true;            forg_f(gi, g[i]){                forg_f_init(g[i]);                if (t != p)ds(i, t);            }        })(-1, 0);        vector<edge_<T>> es;        fora(e, g.edges) { if (can[e.f])es += e; }        rep(i, g.n) {            bool upd = false;            fora(e, es) {                if (res[e.f] != linf && res[e.t] > res[e.f] - e.c) {/*-c*/                upd = true;                    res[e.t] = res[e.f] - e.c;/*-c*/            }            }            if (!upd)break;        }        rep(i, g.n * 2) {            bool upd = 0;            fora(e, g.edges) {                if (res[e.f] != linf && res[e.t] != -linf && res[e.t] > res[e.f] - e.c) {/*-c*/                upd = 1;                    res[e.t] = -linf;                }            }            if (!upd)break;        }        rep(i, g.n)res[i] *= -1;        return res;    }
//コストが負の場合も扱えたはず
template<class T, class I> vector<vector<T>> warshall(graph<T> &g, I init_value) {    int n = g.n;    assert(n < 1e4);    vector<vector<T> > dis(n, vector<T>(n, INF<T>()));    rep(i, n)fora(e, g.g[i]) { if (dis[e.f][e.t] > e.c) { dis[e.f][e.t] = e.c; }}    rep(i, n)dis[i][i] = 0;    rep(k, n) rep(i, n) rep(j, n) { if (dis[i][j] > dis[i][k] + dis[k][j]) { dis[i][j] = dis[i][k] + dis[k][j]; }}    rep(i, n)rep(j, n) if (dis[i][j] == linf)dis[i][j] = init_value;    return dis;}
//密グラフの時、warshallに投げる
//01等の判定もここで出来る
template<class T, class I> vector<vector<T>> dijkstra_all(graph<T> &g, I init_value) {    int n = g.n;    assert(n < 1e4);    if (n * n < (n + sz(g.edges)) * 14) {        /*O(N^3) vs O(N (N+M)log N)*/        return warshall(g, init_value); }    vector<vector<T>> dis(n);    rep(i, n) { dis[i] = dijkstra(g, i, init_value); }    return dis;}
template<class T> class MinOp { public: T operator()(T a, T b) { return min(a, b); }};
template<typename OpFunc> struct SparseTable {    OpFunc op;    signed size;    vector<signed> lg;    vector<vector<pair<signed, signed>>> table;    void init( vector<pair<signed, signed>> &array, OpFunc opfunc) {        signed n = array.size();        op = opfunc;        lg.assign(n + 1, 0);        for (signed i = 1; i <= n; i++) { lg[i] = 31 - __builtin_clz(i); }        table.assign(lg[n] + 1, array);        for (signed i = 1; i <= lg[n]; i++) { for (signed j = 0; j < n; j++) { if (j + (1 << (i - 1)) < n) { table[i][j] = op(table[i - 1][j], table[i - 1][j + (1 << (i - 1))]); } else { table[i][j] = table[i - 1][j]; }}}    }    pair<signed, signed> query(signed l, signed r) {        assert(l < r);        return op(table[lg[r - l]][l], table[lg[r - l]][r - (1 << lg[r - l])]);    }};
struct PMORMQ {    vector<signed> a;    SparseTable<MinOp<pair<signed, signed> > > sparse_table;    vector<vector<vector<signed> > > lookup_table;    vector<signed> block_type;    signed block_size, n_block;    void init( vector<signed> &array) {        a = array;        signed n = a.size();        block_size = std::max(1, (31 - __builtin_clz(n)) / 2);        while (n % block_size != 0) {            a.push_back(a.back() + 1);            n++;        }        n_block = n / block_size;        vector<pair<signed, signed> > b(n_block, make_pair(INT_MAX, INT_MAX));        for (signed i = 0; i < n; i++) { b[i / block_size] = min(b[i / block_size], make_pair(a[i], i)); }        sparse_table.init(b, MinOp<pair<signed, signed> >());        block_type.assign(n_block, 0);        for (signed i = 0; i < n_block; i++) {            signed cur = 0;            for (signed j = 0; j < block_size - 1; j++) {                signed ind = i * block_size + j;                if (a[ind] < a[ind + 1]) { cur |= 1 << j; }            }            block_type[i] = cur;        }        lookup_table.assign(1 << (block_size - 1), vector<vector<signed> >(block_size, vector<signed>(block_size + 1)));        for (signed i = 0; i < (1 << (block_size - 1)); i++) {            for (signed j = 0; j < block_size; j++) {                signed res = 0;                signed cur = 0;                signed pos = j;                for (signed k = j + 1; k <= block_size; k++) {                    lookup_table[i][j][k] = pos;                    if (i & (1 << (k - 1))) { cur++; } else { cur--; }                    if (res > cur) {                        pos = k;                        res = cur;                    }                }            }        }    }    signed query(signed l, signed r) {        assert(l < r);        signed lb = l / block_size;        signed rb = r / block_size;        if (lb == rb) { return lb * block_size + lookup_table[block_type[lb]][l % block_size][r % block_size]; }        signed pl = lb * block_size + lookup_table[block_type[lb]][l % block_size][block_size];        signed pr = rb * block_size + lookup_table[block_type[rb]][0][r % block_size];        signed pos = pl;        if (r % block_size > 0 && a[pl] > a[pr]) { pos = pr; }        if (lb + 1 == rb) { return pos; }        signed spv = sparse_table.query(lb + 1, rb).second;        if (a[pos] > a[spv]) { return spv; }        return pos;    }};

template<class T=int> class tree : public undigraph<T> {
    PMORMQ rmq;    int cnt;    vector<signed> id_, in;    bool never = true;    bool never_hld = true;
    void dfs(int x, int p, int d, int dis = 0) {        id_[cnt] = x;        par_[x] = p;        rmq_dep.push_back(d);        disv[x] = dis;        in[x] = cnt++;        forg_f(gi, g[x]) {            forg_f_init(g[x]);            if (t == p) { continue; }            dfs(t, x, d + 1, dis + c);            id_[cnt] = x;            rmq_dep.push_back(d);            cnt++;        }    }    void precalc() {        never = false;        cnt = 0;        rmq_dep.clear();        disv.assign(n, 0);        in.assign(n, -1);        id_.assign(2 * n - 1, -1);        par_.assign(n, -1);        dfs(root, -1, 0);        rmq.init(rmq_dep);
#ifdef _DEBUG
        if (n >= 100)return;        cerr << "---tree---" << endl;        rep(i, n) {            if (!(i == root || sz(g[i]) > 1))continue;            cerr << i << " -> ";            vi ts;            forg_f(gi, g[i]) {                forg_f_init(g[i]);                if (t != par_[i])ts.push_back(t);            }            rep(i, sz(ts) - 1)cerr << ts[i] << ", ";            if (sz(ts))cerr << ts.back() << endl;        }        cerr << endl;
#endif
    }    int pos;    void hld_build() {        never_hld = false;        if (never)precalc();        g.resize(n);        vid.resize(n, -1);        head.resize(n);        heavy.resize(n, -1);        depth.resize(n);        inv.resize(n);        subl.resize(n);        subr.resize(n);        dfs(root, -1);        t = 0;        dfs_hld(root);
#ifdef _DEBUG
        if (n >= 100)return;        cerr << "---hld_index---" << endl;        vi inds;        rep(i, n) if (sz(g[i]))inds.push_back(i);        rep(i, sz(inds)) {            str s = tos(bel(inds[i]));            cerr << std::right << std::setw(sz(s) + (i ? 1 : 0)) << inds[i];        }        cerr << endl;        rep(i, sz(inds)) { cerr << bel(inds[i]) << " "; }        cerr << endl << endl;        cerr << "---hld_edge_index---" << endl;        fora(e, edges) { if (e.f <= e.t) cerr << e.f << "-" << e.t << " " << bel(e) << endl; }        cerr << endl << endl;        cerr << "!!query!! edge or not edge carefull!!" << endl;
#endif
    }    int dfs(int curr, int prev) {        int sub = 1, max_sub = 0;        heavy[curr] = -1;
        forg_f(gi, g[curr]) {            forg_f_init(g[curr]);            int next = t;            if (next != prev) {                depth[next] = depth[curr] + 1;                int sub_next = dfs(next, curr);                sub += sub_next;                if (max_sub < sub_next) max_sub = sub_next, heavy[curr] = next;            }        }        return sub;    }    int t = 0;
#ifndef __CLION_IDE__
    void dfs_hld(int v = 0) {        vid[v] = subl[v] = t;        t++;        inv[subl[v]] = v;        if (0 <= heavy[v]) {            head[heavy[v]] = head[v];            dfs_hld(heavy[v]);        }
    forg_f(gi, g[v]){forg_f_init(g[v]); if (depth[v] < depth[t])                if (t != heavy[v]) {                    head[t] = t;                    dfs_hld(t);                }        subr[v] = t;    }}
#endif//__CLION_IDE__
    vector<signed> rmq_dep;
    vi par_, depth, disv;
    vi childs;
    vi par_id_;//親への辺のidを持つ
    vvi(ids_);//隣接で辺のidを持つ
public:
    using undigraph<T>::g;    using undigraph<T>::n;    using undigraph<T>::edges;
    int root;
    //(f,t)と(t,f)は同じidを持ち、[0,n-1]でadd順に割り振られる

    //部分木の  [左端、右端)  index
    //部分木の辺に加算する場合
    //add(subl[i],subr[i],x)
    //add(sub[i],sub[i+1],-x)
    vector<int> vid, head, heavy, inv, subl, subr;

    tree(int n_, int root = 0) : undigraph<T>(n_), root(root) { n = n_; }
    tree(int n_, int root, const vector<edge_<T>> &E) : undigraph<T>(n_), root(root) {n = n_;fora(e, E) { add(e.f, e.t, e.c); }}
    // f < t
    void add(int f, int t, T c = 1) {
#ifdef _DEBUG
        if ((n % k5) == 0) {re("tree must resize");}
#endif
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("tree add");            deb(f, t, c);            ole();        }
        g[f].emplace_back(f, t, c);
        g[t].emplace_back(t, f, c);
        edges.emplace_back(f, t, c);/*edgesを使わないなら消せる*/
        edges.emplace_back(t, f, c);
    }
    void add_di(int f, int t, T c = 1) {
#ifdef _DEBUG
        if ((n % k5) == 0) {            re("tree must resize");        }
#endif
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("tree add");            deb(f, t, c);            ole();        }        g[f].emplace_back(f, t, c);        edges.emplace_back(f, t, c);/*edgesを使わないなら消せる*/    }
    void root_change(int roo) {        root = roo;        precalc();    }
    //葉になりえない頂点を根にする
    void root_change_mid() {
#ifdef _DEBUG
        message += "N>2 (mid)\n";
#endif
        assert(n > 2);        rep(i, n) {            if (sz(g[i]) > 1) {                root_change(i);                return;            }        }    }
    int lca(int a, int b) {        if (never)precalc();        int x = in[a];        int y = in[b];        if (x > y) { swap(x, y); }        int pos = rmq.query(x, y + 1);        return id_[pos];    }
    int dis(int a, int b) {        if (never)precalc();        int x = in[a];        int y = in[b];        if (x > y) { swap(x, y); }        int pos = rmq.query(x, y + 1);        int p = id_[pos];        return disv[a] + disv[b] - disv[p] * 2;    }
    int dep(int a) {        if (never)precalc();        return disv[a];    }
    int par(int a) {        if (never)precalc();        return par_[a];    }
    bool isleaf(int i) {if (never)precalc();return (sz(g[i]) == 1 && i != root);}
    bool leaf(int i) { return isleaf(i); }
    vi child(int r) {        vi res;        res.push_back(r);        queue<int> q;        q.push(r);        while (!q.empty()) {            int i = q.front();            res.push_back(i);            q.pop();            forg_f(gi, g[i]) { forg_f_init(g[i]); if (t != par(i))q.push(t); }        }        return res;    }
    vb child_ex(int r) {        vb res(n);        res[r] = true;        queue<int> q;        q.push(r);        while (!q.empty()) {            int i = q.front();            res[i] = true;            q.pop();            forg_f(gi, g[i]) {forg_f_init(g[i]); if (t != par(i))q.push(t); }        }        return res;    }
    int dfs_count_subtree(int p, int i) {        childs[i] = 1;        fort_f(gi, g[i]) {            fort_f_init(gi, g[i]);            childs[i] += dfs_count_subtree(i, t);        }        return childs[i];    }
#define count_child count_subtree
    int count_subtree(int f) {        if (sz(childs) == 0) {            if (never)precalc();            childs.resize(n);            dfs_count_subtree(-1, root);        }        return childs[f];    }
    //fからtの辺を切った時のtの大きさ
    int count_subtree(int f, int t) {        if (par(f) == t) {            return n - count_subtree(f);        } else {            return count_subtree(t);        }    }
    int size(int f, int t){return count_subtree(f, t);}
    int size(){return  n;}
    vi path(int a, int b) {        vi res;        for_each_l(a, b, [&](int i) { res.push_back(i); });        return res;    }
    //idはedgesに使えない ↓edge(id)とする
    //pathに含まれる辺のid達を返す
    vi pathe(int a,int b){
        #ifdef _DEBUG
        static bool was = 0;if(!was)message+="can't edges[id]. must edge(id)\n";was=1;
        #endif
        if(sz(par_id_)==0){            ids_.resize(n);            par_id_.resize(n);            rep(i,0,sz(edges),2){                ids_[edges[i].f].emplace_back(i>>1);                ids_[edges[i].t].emplace_back(i>>1);            }            if(never)precalc();/*par_を呼ぶため*/            rep(i, n){                int pa = par_[i];                forg_f(gi, g[i]){ forg_f_init(g[i]);                   if(t==pa){                        par_id_[i] = ids_[i][gi];                    }                }            }            }        int u = lca(a,b);        vi res;        if (a != u) {            do {                res.emplace_back(par_id_[a]);                a = par_[a];            } while (a != u);        }        vi rev;        if (b != u) {            do {                rev.emplace_back(par_id_[b]);                b = par_[b];            } while (b != u);        }        rer(i,sz(rev)-1){            res.emplace_back(rev[i]);        }        return res;
    }
    //親の辺のidを返す
    int par_id(int i){if(sz(par_id_)==0){pathe(0,0);}return par_id_[i];}
    //fのi番目の辺のidを返す
    int ids(int f,int i) {if(sz(ids_)==0){pathe(0,0);}return ids_[f][i];}
    //idから辺を返す
    edge_<T> edge(int id){return edges[id<<1];}

    /*O(N) hldを使わず木を普通にたどる liteの意*/
    void for_each_l(int u, int v, function<void(int)> fnode) {        int r = lca(u, v);        while (u != r) {            fnode(u);            u = par_[u];        }        fnode(r);        vi a;        while (v != r) {            a.push_back(v);            v = par_[v];        }        while(sz(a)){            fnode(a.back());            a.pop_back();        }    }
    void for_each_edge_l(int u, int v, function<void(edge_<int> &)> fedge) {        int r = lca(u, v);        while (u != r) {                forg_f(gi, g[u]) {forg_f_init(g[u]);                    if (t == par_[u]) {                        fedge(g[u][gi]);                        u = par_[u];                        break;                    }                }            }        vector<edge_<int>> qs;        while (v != r) {                forg_f(gi, g[v]) {  forg_f_init(g[v]);                  if (t == par_[v]) {                        qs.push_back(g[v][gi]);                        v = par_[v];                        break;                    }                }            }        while(sz(qs)){            fedge(qs.back());            qs.pop_back();        }    }
    //Fは半開 (u,v)は木の頂点
    //中ではhldの頂点を見るため、seg木のupdateはhldのindexで行なう
    void for_each_/*[l,r)*/(int u, int v, const function<void(int, int)> &f) {        if (never_hld)hld_build();        while (1) {            if (vid[u] > vid[v]) swap(u, v);            int l = max(vid[head[v]], vid[u]);            int r = vid[v] + 1;            f(l, r);            if (head[u] != head[v]) v = par_[head[v]]; else break;        }    }
    void for_each_edge/*[l,r) O(log(N)) 辺を頂点として扱っている 上と同じ感じで使える*/(int u, int v, const function<void(int, int)> &f) {        if (never_hld)hld_build();        while (1) {            if (vid[u] > vid[v]) swap(u, v);            if (head[u] != head[v]) {                int l = vid[head[v]];                int r = vid[v] + 1;                f(l, r);                v = par_[head[v]];            } else {                if (u != v) {                    int l = vid[u] + 1;                    int r = vid[v] + 1;                    f(l, r);                }                break;            }        }    }
    int bel(int v) {        /*hld内での頂点番号を返す*/        if (never_hld)hld_build();return vid[v];}
    //下の頂点に辺のクエリを持たせている
    int bel(int f, int t) {        /*辺のクエリを扱うときどの頂点に持たせればいいか(vidを返すのでそのままupd出来る)*/        if (never_hld)hld_build();return depth[f] > depth[t] ? vid[f] : vid[t];}
    int bel(edge_<T> &e) {        /*辺のクエリを扱うときどの頂点に持たせればいいか(vidを返すのでそのままupd出来る)*/        if (never_hld)hld_build();return depth[e.f] > depth[e.t] ? vid[e.f] : vid[e.t];}
    template<class ... U> int operator()(U ... args) { return bel(args...); }
    //path l -> r += v
    template<class S> void seg_add(S &seg, int lhei, int rhei, int v) {for_each_(lhei, rhei, [&](int l, int r) { seg.add(l, r, v); });}
    template<class S> void seg_update(S &seg, int lhei, int rhei, int v) {for_each_(lhei, rhei, [&](int l, int r) { seg.update(l, r, v); });}
    template<class S> T seg_get(S &seg, int lhei, int rhei) {        T res = seg.e;        for_each_(lhei, rhei, [&](int l, int r) { res = seg.f(res, seg.get(l, r)); });        return res;    }
    template<class S> void seg_add_edge(S &seg, int lhei, int rhei, int v) { for_each_edge(lhei, rhei, [&](int l, int r) { seg.add(l, r, v); }); }
    template<class S> void seg_update_edge(S &seg, int lhei, int rhei, int v) { for_each_edge(lhei, rhei, [&](int l, int r) { seg.update(l, r, v); }); }
    template<class S> T seg_get_edge(S &seg, int lhei, int rhei) {        T res = seg.e;        for_each_edge(lhei, rhei, [&](int l, int r) { res = seg.f(res, seg.get(l, r)); });        return res;    }
    //単体   edgeは上で処理できる
    template<class S> void seg_add(S &seg, int i, int v) { seg.add(bel(i), v); }
    template<class S> void seg_update(S &seg, int i, int v) { seg.update(bel(i), v); }
    template<class S> T seg_get(S &seg, int i) { return seg[i]; }
    template<class S> void seg_del(S &seg, int i) { seg.del(bel(i)); }

    //部分木iに対するクエリ
    template<class S> void seg_add_sub(S &seg, int i, int v) {if (never_hld)hld_build();seg.add(subl[i], subr[i], v);}
    template<class S> void seg_update_sub(S &seg, int i, int v) {if (never_hld)hld_build();seg.update(subl[i], subr[i], v);}
    template<class S> T seg_get_sub(S &seg, int i, int v) {if (never_hld)hld_build();return seg.get(subl[i], subr[i], v);}
    template<class S> void seg_add_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/seg.add(subl[i] + 1, subr[i], v);}
    template<class S> void seg_update_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/seg.update(subl[i] + 1, subr[i], v);}
    template<class S> T seg_get_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/return seg.get(subl[i] + 1, subr[i], v);}
};
//辺が多いのでedgesを持たない
//cost oo, ox, xo, xx 渡す
template<class T=int> class ungrid_k6 : public undigraph<T> {
    vi costs;
public:
    using undigraph<T>::g;    using undigraph<T>::n;    using undigraph<T>::edges;
    int H, W;
    vector<vector<char>> ba;
    char wall;

    void add(int f, int t, T c = 1){
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("grid_k6 add");            deb(f, t, c);            ole();        }
        g[f].emplace_back(f, t, c);
        g[t].emplace_back(t, f, c);
        //undigraphと違い、edgesを持たない
    }
    int getid(int h, int w) {        assert(ins(h, w, H, W));        return W * h + w;    }
    int getid(P p) { return getid(p.first, p.second); }
    P get2(int id) { return mp(id / W, id % W); }
    P operator()(int id) { return get2(id); }
    int operator()(int h, int w) { return getid(h, w); }
    int operator()(P p) { return getid(p); }
    //辺は無い
    ungrid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {}
    //fh,fw,th,twでコストを返す linfならスルー
    template<class F>void build(F func) {            rep(h, H) {                rep(w, W) {                    int fid = getid(h, w);                    auto add = [&](int fh, int fw, int th, int tw) {                        if (ins(th, tw, H, W)) {                            auto cost = func(fh, fw, th, tw);                            if (abs(cost) >= linf)return;                            g[fid].emplace_back(fid, getid(th, tw), cost);                        }                    };                    add(h, w, h + 1, w);                    add(h, w, h - 1, w);                    add(h, w, h, w + 1);                    add(h, w, h, w - 1);                }            }        }
    ungrid_k6(vector<vector<char>> ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])) {            build([&](int fh, int fw, int th, int tw) {                if (ba[fh][fw] == wall || ba[th][tw] == wall)return linf;                else return 1ll;            });        }
    /*o -> o, o -> x, x -> o, x-> x*/
    ungrid_k6(vector<vector<char>> ba, int oo, int ox, int xo, int xx, char wall = '#') : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])), costs({oo, ox, xo, xx}), ba(ba), wall(wall) {            build([&](int fh, int fw, int th, int tw) {                int cm = 0;                if (ba[fh][fw] == wall) { cm += 2; }                if (ba[th][tw] == wall) { cm++; }                return costs[cm];            });        }

    //lambda
    ungrid_k6(vector<vector<char>> ba, char wall, function<T(char, char)> f) : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])), ba(ba), wall(wall) {            build([&](int fh, int fw, int th, int tw) {                return f(ba[fh][fw], ba[th][tw]);            });        }
    ungrid_k6(vector<vector<char>> ba, char wall, function<T(int, int, int, int)> f) : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])), ba(ba), wall(wall) {            build(f);        }
    void set_edges() { rep(i, n)fora(e, g[i])if (e.f < e.t)edges.push_back(e); }
};

//辺によりメモリを大量消費ためedgesを消している
//頂点10^6でメモリを190MB(制限の8割)使う
//左上から右下に移動できる
template<class T=int> class digrid_k6 : public digraph<T> {
    vi costs;
public:
     vector<vector<char>> ba;
    char wall;
   using digraph<T>::g;    using digraph<T>::n;    using digraph<T>::edges;
    int H, W;
    void add(int f, int t, T c = 1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("digrid_k6 add");            deb(f, t, c);            ole();        }
        g[f].emplace_back(f, t, c);
        /*digraphと違いedgesを持たない*/
    }
    int getid(int h, int w) {        if (!ins(h, w, H, W))return -1;        return W * h + w;    }
    P get2(int id) { return mp(id / W, id % W); }
    P operator()(int id) { return get2(id); }
    int operator()(int h, int w) { return getid(h, w); }
    digrid_k6(int H, int W) : H(H), W(W), digraph<T>(H * W){}
    //fh,fw,th,twでコストを返す linfならスルー
    template<class F>void build(F func) {            rep(h, H) {                rep(w, W) {                    int fid = getid(h, w);                    auto add = [&](int fh, int fw, int th, int tw) {                        if (ins(th, tw, H, W)) {                            auto cost = func(fh, fw, th, tw);                            if (abs(cost) >= linf)return;                            g[fid].emplace_back(fid, getid(th, tw), cost);                        }                    };                    add(h, w, h + 1, w);                    add(h, w, h, w + 1);                }            }        }
    digrid_k6(vector<vector<char>> ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])) {            build([&](int fh, int fw, int th, int tw) {                if (ba[fh][fw] == wall || ba[th][tw] == wall)return linf;                else return 1ll;            });        }
    /*o -> o, o -> x, x -> o, x-> x*/
    digrid_k6(vector<vector<char>> ba, int oo, int ox, int xo, int xx, char wall = '#') : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])), costs({oo, ox, xo, xx}), ba(ba), wall(wall) {            build([&](int fh, int fw, int th, int tw) {                int cm = 0;                if (ba[fh][fw] == wall) { cm += 2; }                if (ba[th][tw] == wall) { cm++; }                return costs[cm];            });        }

    //lambda
    digrid_k6(vector<vector<char>> ba, char wall, function<T(char, char)> f) : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])), ba(ba), wall(wall) {            build([&](int fh, int fw, int th, int tw) {                return f(ba[fh][fw], ba[th][tw]);            });        }
    digrid_k6(vector<vector<char>> ba, char wall, function<T(int, int, int, int)> f) : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])), ba(ba), wall(wall) {            build(f);        }
    void add(int fh, int fw, int th, int tw) { add(getid(fh, fw), getid(th, tw)); }
    void set_edges() { rep(i, n)fora(e, g[i])edges.push_back(e); }
};
//grid用のdijkstra
//vector<vector<t>>を返す
template<class G, class I> auto dijkstra(G &g, int fh, int fw, I init_value) {    auto dis = dijkstra(g, g(fh, fw), init_value);    vvi(res, g.H, g.W);    rep(i, sz(dis)) {        int h, w;        tie(h, w) = g.get2(i);        res[h][w] = dis[i];    }    return res;}

//edgesを持たない
//dijkstra(g,0)[t] とかける (t+n-1等とする必要はない)
template<class T=int>class segdi : public digraph<T>{int getid(int k){        if(k >= len*2-1){            return k;        }        else if(k < len - 1){            return k + len;        }else{            return k - len + 1;        }    }void add(int f, int t, T c = 1) {        f = getid(f);        t = getid(t);        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("segdi add");            deb(f, t, c);            ole();        }        g[f].emplace_back(f, t, c);        /*digraphと違いedgesを持たない*/    }int mid;int len;
public:
    using digraph<T>::g;    using digraph<T>::n;    using digraph<T>::edges;
    //頂点が足りなくなったらresize
    segdi(int n_) : digraph<T>(1) {        int nn = 1;        while (nn < n_)nn *= 2;        n_ = nn;        len = n_;        this->resize(n_ + (n_ - 1) * 2 + n_);        mid = n_ + (n_ - 1) * 2;        int ad = len * 2 - 1;        rep(i, len - 1) {            add(i, i * 2 + 1, 0);            add(i, i * 2 + 2, 0);            if (i * 2 + 1 >= len - 1) {                add(i * 2 + 1, i + ad, 0);                add(i * 2 + 2, i + ad, 0);            } else {                add(i * 2 + 1 + ad, i + ad, 0);                add(i * 2 + 2 + ad, i + ad, 0);            }        }    }
    void dfs(vi &list, int nl, int nr, int k, int l, int r) {        if (r <= nl || nr <= l)return;        if (l <= nl && nr <= r) {            list += k;        } else {            dfs(list, nl, (nl + nr) / 2, k * 2 + 1, l, r);            dfs(list, (nl + nr) / 2, nr, k * 2 + 2, l, r);        }    }
    void rekkyo(vi &list, int l, int r) {        l += len - 1;        r += len - 1;        while (l < r) {            if (!(l & 1)) { list.push_back(l); }            if (!(r & 1)) { list.push_back(r - 1); }            l >>= 1;            r = (r - 1) >> 1;        }    }
    //半開
    void add(int fl, int fr, int tl, int tr, int cost) {        /*fは下側*/        int ad = 2 * len - 1;        if (mid >= n) { this->resize(n + 100); }        {            int l = fl, r = fr;            l += len - 1;            r += len - 1;            while (l < r) {                if (!(l & 1)) {                    if (l - len + 1 < 0)add(l + ad, mid, cost);                    else add(l, mid, cost);                }                if (!(r & 1)) {                    if (r - 1 - len + 1 < 0)add(r - 1 + ad, mid, cost);                    else add(r - 1, mid, cost);                }                l >>= 1;                r = (r - 1) >> 1;            }        }        {            int l = tl, r = tr;            l += len - 1;            r += len - 1;            while (l < r) {                if (!(l & 1)) { add(mid, l, 0); }                if (!(r & 1)) { add(mid, r - 1, 0); }                l >>= 1;                r = (r - 1) >> 1;            }        }        mid++;    }
};
//edgesを持たない
//dijkstra(g,0)[t] とかける (t+n-1等とする必要はない)
template<class T=int>class segun : public undigraph<T>{int getid(int k){        if(k >= len*2-1){            return k;        }        else if(k < len - 1){            return k + len;        }else{            return k - len + 1;        }    }void add(int f, int t, T c = 1) {        f = getid(f);        t = getid(t);        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("segun add");            deb(f, t, c);            ole();        }        g[f].emplace_back(f, t, c);        /*digraphと違いedgesを持たない*/            g[t].emplace_back(t, f, c);        /*digraphと違いedgesを持たない*/    }    int mid;    int len;
public:
    using digraph<T>::g;    using digraph<T>::n;    using digraph<T>::edges;
    //頂点が足りなくなったらresize
    segun(int n_) : digraph<T>(1) {        int nn = 1;        while (nn < n_)nn *= 2;        n_ = nn;        len = n_;        this->resize(n_ + (n_ - 1) * 2 + n_);        mid = n_ + (n_ - 1) * 2;        int ad = len * 2 - 1;        rep(i, len - 1) {            add(i, i * 2 + 1, 0);            add(i, i * 2 + 2, 0);            if (i * 2 + 1 >= len - 1) {                add(i * 2 + 1, i + ad, 0);                add(i * 2 + 2, i + ad, 0);            } else {                add(i * 2 + 1 + ad, i + ad, 0);                add(i * 2 + 2 + ad, i + ad, 0);            }        }    }
    void dfs(vi &list, int nl, int nr, int k, int l, int r) {        if (r <= nl || nr <= l)return;        if (l <= nl && nr <= r) {            list += k;        } else {            dfs(list, nl, (nl + nr) / 2, k * 2 + 1, l, r);            dfs(list, (nl + nr) / 2, nr, k * 2 + 2, l, r);        }    }
    void rekkyo(vi &list, int l, int r) {        l += len - 1;        r += len - 1;        while (l < r) {            if (!(l & 1)) { list.push_back(l); }            if (!(r & 1)) { list.push_back(r - 1); }            l >>= 1;            r = (r - 1) >> 1;        }    }
    //半開
    void add(int fl, int fr, int tl, int tr, int cost) {        /*fは下側*/        int ad = 2 * len - 1;        if (mid >= n) { this->resize(n + 100); }        {            int l = fl, r = fr;            l += len - 1;            r += len - 1;            while (l < r) {                if (!(l & 1)) {                    if (l - len + 1 < 0)add(l + ad, mid, cost);                    else add(l, mid, cost);                }                if (!(r & 1)) {                    if (r - 1 - len + 1 < 0)add(r - 1 + ad, mid, cost);                    else add(r - 1, mid, cost);                }                l >>= 1;                r = (r - 1) >> 1;            }        }        {            int l = tl, r = tr;            l += len - 1;            r += len - 1;            while (l < r) {                if (!(l & 1)) { add(mid, l, 0); }                if (!(r & 1)) { add(mid, r - 1, 0); }                l >>= 1;                r = (r - 1) >> 1;            }        }        mid++;    }
};
#define getid_2(h, w) ((h) * (W) + (w))
#define getid_1(p) ((p).first * W + (p).second)
#define o_getid(a,b,name,...) name
#define getid(...) o_getid(__VA_ARGS__, getid_2, getid_1) (__VA_ARGS__)

#define unionfind unionfind_graph
struct unionfind {
    vector<ll> par;
    vector<ll> siz;
    vector<ll> es;
    ll n, trees;//連結グループの数(親の種類)
    unionfind(ll n) : n(n), trees(n) {        par.resize(n);        siz.resize(n);        es.resize(n);        for (ll i = 0; i < n; i++) {            par[i] = i;            siz[i] = 1;        }    }
    template<class T>unionfind(graph<T>& g) : n(g.n), trees(g.n) {        par.resize(n);        siz.resize(n);        es.resize(n);        for (ll i = 0; i < n; i++) {            par[i] = i;            siz[i] = 1;        }        add(g);    }
    ll root(ll x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}
    ll operator()(ll x){return root(x);}
    bool unite(ll x, ll y) {
        x = root(x);
        y = root(y);
        es[x]++;
        if (x == y) return false;
        if (siz[x] > siz[y]) swap(x, y);
        trees--;
        par[x] = y;
        siz[y] += siz[x];
        es[y] += es[x];
        return true;
    }
    template<class T>void add(graph<T>&g){fora(e,g.edges){unite(e.f,e.t);}}
    template<class T>void unite(graph<T>&g){add(g);}
    bool same(ll x, ll y) { return root(x) == root(y); }
    ll size(ll x) { return siz[root(x)]; }
    ll esize(ll x) { return es[root(x)]; }
    vi sizes(){        vi cou(n);        vi ret;        ret.reserve(n);        rep(i, n){            cou[root (i)]++;        }        rep(i, n){            if(cou[i])ret.push_back(cou[i]);        }        return ret;    }
    //つながりを無向グラフと見なし、xが閉路に含まれるか判定
    bool close(ll x) { return esize(x) >= size(x); }
    vector<vi> sets() {        vi ind(n, -1);        ll i = 0;        vvi(res, trees);        rep(j, n) {            ll r = root(j);            if (ind[r] == -1)ind[r] = i++;            res[ind[r]].push_back(j);        }        rep(i, trees) {            ll r = root(res[i][0]);            if (res[i][0] == r)continue;            rep(j, 1, sz(res[i])) {                if (res[i][j] == r) {                    swap(res[i][0], res[i][j]);                    break;                }            }        }        return res;    }
};//@formatter:off
//forg基本

template<class T> struct forg_edge_body {
    vector<edge_<T>> &edges;
    int i;
    explicit forg_edge_body(int i, vector<edge_<T>> &edges): i(i), edges(edges) {}
    auto operator++() {i++;}
    auto operator*() {
        #ifdef type_id_graph
            return tuple<int, int, int, T, int, int>(i, edges[i].f, edges[i].t, edges[i].c, edges[i].ty, edges[i].id);
        #else
            return tuple<int, int, int, T>(i, edges[i].f, edges[i].t, edges[i].c);
        #endif
    }
    template<class U>auto operator!=(const U& rhs){return i != rhs.i;}
};
/*@formatter:on*/

    template<class T> struct fort_edge_body {
        vector<edge_<T>> &edges;
        int gi, ci;
        int par;
        void can_gi() {
            while (gi < sz(edges) && edges[gi].t == par) {
                gi++;
            }
        }
        explicit fort_edge_body(int i, int par, vector<edge_<T>> &edges) : gi(i), ci(i), par(par), edges(edges) {
            can_gi();
        }
        auto operator++() {
            ci++;
            gi++;
            can_gi();
        }
        auto operator*() {
#ifdef type_id_graph
            return tuple<int, int, int, T, int, int>(gi, edges[gi].f, edges[gi].t, edges[gi].c, edges[gi].ty, edges[gi].id);
#else
            return tuple < int, int, int, T > (ci, edges[gi].f, edges[gi].t, edges[gi].c);
#endif
        }
        template<class U> auto operator!=(const U &rhs) { return gi != rhs.gi; }
    };
    template<class T> struct forg_edge {
        vector<edge_<T>> &edges;
        forg_edge_body<T> endp;
        forg_edge(vector<edge_<T>> &edges) : edges(edges), endp(sz(edges), edges) {}
        auto begin() { return forg_edge_body<T>(0, edges); }
        auto end() { return endp; }
    };
    template<class T> struct fort_edge {
        vector<edge_<T>> &edges;
        fort_edge_body<T> endp;
        int p;
        fort_edge(int p, vector<edge_<T>> &edges) : p(p), edges(edges), endp(sz(edges), p, edges) {}
        auto begin() { return fort_edge_body<T>(0, p, edges); }
        auto end() { return endp; }
    };
/*@formatter:off*/
/*@formatter:off*/
#ifdef type_id_graph
    #define forg(gi, edges) for(auto[gi, f, t, c, ty, id] :  forg_edge(edges))
    #define fort(ci, edges) for(auto[ci, f, t, c, ty, id] :  fort_edge(p, edges))
#else
    #define forg(gi, edges) for(auto[gi, f, t, c] :  forg_edge(edges))
    #define fort(ci, edges) for(auto[ci, f, t, c] :  fort_edge(p, edges))
#endif

//@出力
template<class T> ostream &operator<<(ostream &os, digraph<T> &g) {    os << endl << g.n << " " << sz(g.edges) << endl;    fore(gi, g.edges)    { os << f << " " << t << " " << c << endl; }    return os;}template<class T> ostream &operator<<(ostream &os, undigraph<T> &g) {    os << endl << g.n << " " << sz(g.edges) / 2 << endl;    fore(gi, g.edges){ if (f < t)os << f << " " << t << " " << c << endl; }    return os;}
//@判定
template<class T> bool nibu( graph<T> &g) {    int size = 0;    rep(i, g.n)size += sz(g.g[i]);    if (size == 0)return true;    unionfind uf(g.n * 2);    rep(i, g.n)fora(e, g.g[i])uf.unite(e.f, e.t + g.n), uf.unite(e.f + g.n, e.t);    rep(i, g.n)if (uf.same(i, i + g.n))return 0;    return true;}
//頂点ではなく辺の数に依存した計算量 O(E)
template<class T> bool nibu_sub( graph<T> &g) {    umapi col;/*0なら無色 */   queue<int> q;    /*色は01か11 */    fora(e, g.edegs) {        /*fとtの色を持つか否かは同じ*/        if (col[e.f] == 0)q.push(e.f);        while (!q.empty()) {            int f = q.front();            q.pop();            int fc = col[f];            forg_f(gi, g[f]) {  forg_f_init(g[f]);              int &tc = col[t];                /*fcには色がある*/                if (fc == tc)return false;                /*違う色*/                if (tc)continue;                /*無色*/                tc = fc ^ 2;                q.push(tc);            }        }    }    return true;}
    //二部グラフを色分けした際の頂点数を返す
template<class T> vp nibug(graph<T> &g) {    vp cg;    if (!nibu(g)) {        debugline("nibu");        ole();    }    int n = g.size();    vb was(n);    queue<P> q;    rep(i, n) {        if (was[i])continue;        q.push(mp(i, 1));        was[i] = 1;        int red = 0;        int coun = 0;        while (q.size()) {            int now = q.front().fi;            int col = q.front().se;            red += col;            coun++;            q.pop();            forg_f(gi, g[now]) {   forg_f_init(g[now]);             if (was[t])continue;                q.push(mp(t, col ^ 1));                was[t] = 1;            }        }        cg.push_back(mp(red, coun - red));    }    return cg;}
    //連結グラフが与えられる 閉路があるか
template<class T> bool close(undigraph<T> &g) {    int n = 0;    int e = 0;    rep(i, g.n) {        if (sz(g[i]))n++;        forg_f(gi, g[i]) {forg_f_init(g[i]); e++; }    }    return (e >> 1) >= n;}
template<class T> bool close(undigraph<T> &g, int v) {    unionfind uf(g.n);    rep(i, g.n) {        forg_f(gi, g[i]) {forg_f_init(g[i]);            if (f < t)break;            if (f == t && f == v)return true;            if (uf.same(f, v) && uf.same(t, v))return true;            uf.unite(f, t);        }    }    return false;}template<class T> bool close(digraph<T> &g) {    vi res;    return topo(res, g);}

//@変形
//条件(f!=0等 f,t,cが使える)を満たすsubgraphをg2に代入
#define sub_di(g, g2, zhouken) {g2.resize(g.n);fore(gi, g.edges){if(zhouken){g2.add(f,t,c);}}}
#define sub_un(g, g2, zhouken) {g2.resize(g.n);fore(gi, g.edges){bool ok = zhouken; /*片方がアウトなら駄目という扱い*/swap(f,t); ok &= zhouken;if(ok && f < t){g2.add(f,t,c);}}}
//誘導部分グラフ(両辺がVに含まれる辺を含む最大のグラフを返す) yuudou
#define sub_di_guided(g, g2, V) {vi ex(max(V)); fora(v,V){ex[v]=1;}sub_di(g, g2, ex[f] && ex[t]);}
#define sub_un_guided(g, g2, V) {vi ex(max(V)); fora(v,V){ex[v]=1;}sub_un(g, g2, ex[f] && ex[t]);}

#define sub_tree sub_un
    //閉路がなければtrue
bool topo(vi &res, digraph<int> &g) {    int n = g.g.size();    vi nyu(n);    rep(i, n)for (auto &&e :g[i])nyu[e.t]++;    queue<int> st;    rep(i, n)if (nyu[i] == 0)st.push(i);    while (st.size()) {        int v = st.front();        st.pop();        res.push_back(v);        fora(e, g[v]) if (--nyu[e.t] == 0)st.push(e.t);    }    return res.size() == n;}
    //辞書順最小トポロジカルソート
bool topos(vi &res, digraph<int> &g) {    int n = g.g.size();    vi nyu(n);    rep(i, n)for (auto &&e :g[i])nyu[e.t]++;    /*小さい順*/    priority_queue<int, vector<int>, greater<int> > q;    rep(i, n)if (nyu[i] == 0)q.push(i);    while (q.size()) {        int i = q.top();        q.pop();        res.push_back(i);        fora(e, g[i])if (--nyu[e.t] == 0)q.push(e.t);    }    return res.size() == n;}
template<class T> digraph<T> rev(digraph<T> &g) {    digraph<T> r(g.n);    rep(i, g.n) { forg_f(gi, g[i]) {forg_f_init(g[i]); r.add(t, f, c); }}    return r;}
    //lc,rcは子を持つ中で一番左、右
    //(g,ind,l,r)
template<class T> tree<T> get_bfs_tree(tree<T> &g, vi &ind, vi &l, vi &r) {if (sz(ind)) {cerr << "ind must be empty" << endl;exit(0);}int N = sz(g);ind.resize(N);l.resize(N, inf);r.resize(N, -1);tree<T> h(N);queue<P> q;q.emplace(-1, 0);int c = 0;while (sz(q)) {int p = q.front().first;int i = q.front().second;q.pop();ind[i] = c;if (~p)chmi(l[ind[p]], c);if (~p)chma(r[ind[p]], c);c++;forg_f(gi, g[i]) {forg_f_init(g[i]);if (t != p)q.emplace(i, t);}}fora(e, g.edges) {if (e.f < e.t) {h.add(ind[e.f], ind[e.t], e.c);}}rep(i, N) {if (l[i] == inf)l[i] = -1;}return h;}
    //lc,rcは子を持つ中で一番左、右
    // たとえばl[lc[x]は2段下の最左
    //(g,ind,l,r,lc,rc)
template<class T> tree<T> get_bfs_tree(tree<T> &g, vi &ind, vi &l, vi &r, vi &lc, vi &rc) {    if (sz(ind)) {        cerr << "ind must be empty" << endl;        exit(0);    }    int N = sz(g);    ind.resize(N);    l.resize(N, inf);    lc.resize(N, inf);    r.resize(N, -1);    rc.resize(N, -1);    tree<T> h(N);    queue<P> q;    q.emplace(-1, 0);    int c = 0;    while (sz(q)) {        int p = q.front().first;        int i = q.front().second;        q.pop();        ind[i] = c;        if (~p) {            chmi(l[ind[p]], c);            chma(r[ind[p]], c);            if (sz(g[i]) > 1) {                chmi(lc[ind[p]], c);                chma(rc[ind[p]], c);            }        }        c++;        forg_f(gi, g[i]) {   forg_f_init(g[i]);         if (t != p)q.emplace(i, t);        }    }    fora(e, g.edges) {        if (e.f < e.t) {            h.add(ind[e.f], ind[e.t], e.c);        }    }    rep(i, N) {        if (l[i] == inf)l[i] = -1;        if (lc[i] == inf)lc[i] = -1;    }    return h;}

//@集計
template<class T> vi indegree(graph<T> &g) {vi ret(g.size());rep(i, g.size()) { forg_f(gi, g[i]) {forg_f_init(g[i]); ret[t]++; }}return ret;}
template<class T> vi outdegree(graph<T> &g) {vi ret(g.size());rep(i, g.size()) { ret[i] = g[i].size(); }return ret;}
#define kansetu articulation
//private
/*private*/  P farthest____(tree<> &E, int cur, int pre, int d, vi &D) {    D[cur] = d;    P r = {d, cur};
        forg_f(gi, E[cur]) {
            forg_f_init(E[cur]);
            if (t != pre) {
                P v = farthest____(E, t, cur, d + c, D);
                r = max(r, v);
            }
        }
        return r;
    }
//dagでなければ-1を返す
int diameter(digraph<> &g) {    vi per;    if (!topo(per, g))return -1;    int n = g.n;    vi dp(n);    fora(v, per) { forg_f(gi, g[v]) { forg_f_init(g[v]);chma(dp[t], dp[f] + c); }}    return max(dp);}
//iから最も離れた距離
vi diameters(tree<> &E) { /* diameter,center*/vi D[3];    D[0].resize(E.size());    D[1].resize(E.size());    auto v1 = farthest____(E, 0, 0, 0, D[0]);    auto v2 = farthest____(E, v1.second, v1.second, 0, D[0]);    farthest____(E, v2.second, v2.second, 0, D[1]);    int i;    rep(i, D[0].size()) D[2].push_back(max(D[0][i], D[1][i]));    return D[2];}
//iに対応するjと距離
vp diameters_p(tree<> &E) { /* diameter,center*/vector<int> D[3];    D[0].resize(E.size());    D[1].resize(E.size());    auto v1 = farthest____(E, 0, 0, 0, D[0]);    auto v2 = farthest____(E, v1.second, v1.second, 0, D[0]);    farthest____(E, v2.second, v2.second, 0, D[1]);    int i;    vp res(E.size());    rep(i, D[0].size()) { if (D[0][i] > D[1][i])res[i] = mp(v1.second, D[0][i]); else res[i] = mp(v2.second, D[1][i]); }    return res;}
int diameter(tree<> &E) {    vi d = diameters(E);    return max(d);}
//最も離れた二点を返す
P diameter_p(tree<> &E) {    auto d = diameters_p(E);    int dis = -1;    int l = -1, r = -1;    rep(i, sz(d)) {        if (chma(dis, d[i].se)) {            l = i;            r = d[i].fi;        }    }    return mp(l, r);}


//@列挙 取得
//閉路がある時linfを返す
template<class T> int longest_path(digraph<T> &g) {    vi top;    if (!topo(top, g)) { return linf; }    int n = sz(top);    vi dp(n, 0);    for (auto &&i : top) { forg_f(gi, g[i]) { forg_f_init(g[i]);chma(dp[t], dp[i] + 1); }}    return max(dp);}
template<class T> vi longest_path_v(digraph<T> &g) {    vi top;    if (!topo(top, g)) { return vi(); }    int n = sz(top);    vi dp(n, 0);    vi pre(n, -1);    for (auto &&i : top) { forg_f(gi, g[i]) { forg_f_init(g[i]);if (chma(dp[t], dp[i] + 1)) { pre[t] = i; }}}    int s = std::max_element(dp.begin(), dp.end()) - dp.begin();    vi path;    while (s != -1) {        path.push_back(s);        s = pre[s];    }    std::reverse(path.begin(), path.end());    return path;}
//橋を列挙する (取り除くと連結でなくなる辺)
template<class T> vp bridge(graph<T> &G) {    static bool was;    vp brid;    vi articulation;    vi ord(G.n), low(G.n);    vb vis(G.n);    function<void(int, int, int)> dfs = [&](int v, int p, int k) {        vis[v] = true;        ord[v] = k++;        low[v] = ord[v];        bool isArticulation = false;        int ct = 0;        for (int i = 0; i < G[v].size(); i++) {            if (!vis[G[v][i].t]) {                ct++;                dfs(G[v][i].t, v, k);                low[v] = min(low[v], low[G[v][i].t]);                if (~p && ord[v] <= low[G[v][i].t]) isArticulation = true;                if (ord[v] < low[G[v][i].t]) brid.push_back(make_pair(min(v, G[v][i].t), max(v, G[v][i].t)));            } else if (G[v][i].t != p) { low[v] = min(low[v], ord[G[v][i].t]); }        }        if (p == -1 && ct > 1) isArticulation = true;        if (isArticulation) articulation.push_back(v);    };    int k = 0;    rep(i, G.n) { if (!vis[i]) dfs(i, -1, k); }    sort(brid.begin(), brid.end());    return brid;}
//間接点を列挙する (取り除くと連結でなくなる点)
template<class T> vi articulation(undigraph<T> &G) {    static bool was;    vp bridge;    vi arti;    vi ord(G.n), low(G.n);    vb vis(G.n);    function<void(int, int, int)> dfs = [&](int v, int p, int k) {        vis[v] = true;        ord[v] = k++;        low[v] = ord[v];        bool isArticulation = false;        int ct = 0;        for (int i = 0; i < G[v].size(); i++) {            if (!vis[G[v][i].t]) {                ct++;                dfs(G[v][i].t, v, k);                low[v] = min(low[v], low[G[v][i].t]);                if (~p && ord[v] <= low[G[v][i].t]) isArticulation = true;                if (ord[v] < low[G[v][i].t]) bridge.push_back(make_pair(min(v, G[v][i].t), max(v, G[v][i].t)));            } else if (G[v][i].t != p) { low[v] = min(low[v], ord[G[v][i].t]); }        }        if (p == -1 && ct > 1) isArticulation = true;        if (isArticulation) arti.push_back(v);    };    int k = 0;    rep(i, G.n) { if (!vis[i]) dfs(i, -1, k); }    sort(arti.begin(), arti.end());    return arti;}
//閉路パスを一つ返す

vi close_path(digraph<> &g) {    int n = g.n;    vi state(n);    vi path;    rep(i, n) if (!state[i]) {            if (fix([&](auto dfs, int v) -> bool {                if (state[v]) {                    if (state[v] == 1) {                        path.erase(path.begin(), find(path.begin(), path.end(), v));                        return true;                    }                    return false;                }                path.push_back(v);                state[v] = 1;                forg_f(gi, g[v]) { forg_f_init(g[i]);if (dfs(t))return true; }                state[v] = -1;                path.pop_back();                return false;            })(i)) { return path; }        }    return vi();}
#ifdef type_id_graph
vi close_path_id(digraph<> &g) {    auto D = close_path(g);    if(sz(D)==0)return vi();    D += D[0];    vi ret;    int cur = D[0];    rep(i,sz(D)-1){        forg_f(gi, g[D[i]]){forg_f_init(g[D[i]]);            if(t == D[i+1]){                ret += id;                break;            }        }    }    return ret;}
#endif
vi close_path_min(digraph<> &g) {    int n = g.n;    vvi(dis, n);    rep(i, n)dis[i] = dijkstra(g, i, linf);    int mind = linf;    int f = 0, t = 0;    rep(i, n) {        rep(j, n) {            if (i == j)continue;            if (chmi(mind, dis[i][j] + dis[j][i])) {                f = i;                t = j;            }        }    }    vi path;    auto add = [&](int f, int t) {        int now = f;        while (now != t) {            rep(i, n) {                if (dis[now][i] == 1 && dis[f][i] + dis[i][t] == dis[f][t]) {                    path.push_back(i);                    now = i;                    break;                }            }        }    };    add(f, t);    add(t, f);    return path;}
//iを含む最短閉路 https://atcoder.jp/contests/abc022/tasks/abc022_c
/*閉路が1つしかない場合、その閉路に含まれる頂点を1としたvectorを返す*/;
template<class T> vi get_close1(digraph<T> &g) {    int n = g.n;    queue<int> q;    vi d = outdegree(g);    vi res(n, 1);    rep(i, n) {        if (d[i] == 0) {            q += i;            res[i] = 0;        }    }    auto rg = rev(g);    while (q.size()) {        auto now = q.front();        q.pop();        forg_f(gi, rg[now]) {    forg_f_init(rg[now]);        if (--d[t] == 0) {                q += t;                res[t] = 0;            }        }    }    return res;}

//@アルゴリズム
//下にkrus_idがある
#define mst krus
template<class T> vi krus_i(vector<edge_<T>> &g, int group = 1, int n = -1) {    if (n == -1)fora(e, g) { chma(n, max(e.f, e.t) + 1); };    vi res;    unionfind uf(n);    auto eis = sorti(g);    int gcou = n;    fora(ei, eis) {        if (gcou == group)break;        if (uf.unite(g[ei].f, g[ei].t)) {            res += ei;            gcou--;        }    }    return res;}
template<class T> vector<edge_<T>> krus_ed(vector<edge_<T>> &g, int group=1, int n = -1) {    auto inds = krus_i(g, group, n);    vector<edge_<T>> ret;    fora(i, inds) ret += g[i];    return ret;}
template<class T> vector<edge_<T>> krus_ed(undigraph<T> &g, int group = 1) {    if (sz(g.edges) == 0)g.set_edges();    return krus_ed(g.edges, group, g.n);}
template<class T> T krus(undigraph<T> &g, int group = 1) {    auto edges = krus_ed(g, group);    T res=0;    fora(e,edges)res += e.c;    return res;}
//森を返す
template<class T> undigraph<T> krus_un(undigraph<T> &g, int group/*森になるので*/) {    return undigraph<T>(g.n, krus_ed(g, group));}
template<class T> tree<T> krus_tr(undigraph<T> &g) {    return tree<T>(g.n, 0,krus_ed(g));}

template<class T> int krus(vector<edge_<T>> &g, int group = 1) {    auto edges = krus_ed(g, group);    T res=0;    fora(e, edges)res += e.c;    return res;}


/*@formatter:off*/
//@実験
digraph<> rang_di(int n, int m, bool zibun = 0, bool taju = 0) {umapp was;    digraph<> g(n);    was[mp(-1, -2)] = 1;    while (m) {        int f = -1, t = -2;        while (f < 0 || (!taju && was[mp(f, t)])) {            f = rand(0, n - 1);            t = rand(0, n - 1);            if (!zibun && f == t)f = -1;        }        g.add(f, t);        was[mp(f, t)] = 1;        m--;    }    return g;}
digraph<> perfect_di(int n, bool zibun = 0) {    digraph<> g(n);    rep(i, n) {        rep(j, n) {            if (!zibun && i == j)con;            g.add(i, j);        }    }    return g;}
undigraph<> rang_un(int n, int m, bool zibun = 0, bool taju = 0) {    umapp was;    undigraph<> g(n);    was[mp(-1, -2)] = 1;    while (m) {        int f = -1, t = -2;        while (f < 0 || (!taju && was[mp(min(f, t), max(f, t))])) {            f = rand(0, n - 1);            t = rand(0, n - 1);            if (!zibun && f == t)f = -1;        }        g.add(f, t);        was[mp(min(f, t), max(f, t))] = 1;        m--;    }    return g;}
undigraph<> perfect_un(int n, bool zibun = 0){    undigraph<> g(n);    rep(i, n) {        rep(j, i, n) {            if (!zibun && i == j)con;            g.add(i, j);        }    }    return g;}
/*頂点数がkの木を一つ返す サイズが0の木が帰ったら終了*/
tree<int> next_tree(int k) {    assert(2 <= k && k < 11);    static str name;    static ifstream ina;    static int rem;    static vp edges;    static int pk = -1;/*前回見たk*/    if (pk != k) {        if (~pk)ina.close();        edges.clear();        pk = k;        name = (k == 6) ? "C:\\Users\\kaout\\Desktop\\trees_sizek\\nazeka6.txt" : "C:\\Users\\kaout\\Desktop\\trees_sizek\\tree_size" + tos(k) + ".txt";        ina = ifstream(name);        rem = pow(k, k - 2);/*Cayleyの定理*/        rep(i, k)rep(j, i + 1, k)edges.emplace_back(i, j);        pk = k;    }    tree<int> g(k);    if (rem == 0) {        g.resize(0);        return g;    }    int m;    ina >> m;    while (m) {        int lb = lbit(m);        int id = log2(lb);        g.add(edges[id].first, edges[id].second);        m ^= lb;    }    rem--;    return g;}
undigraph<int> next_undi(int k) {    assert(2 <= k && k < 9);    static str name;    static ifstream ina;    static int rem;    static vp edges;    static vi lims = {-1, -1, 1, 4, 38, 728, 26704, 1866256};    static int pk = -1;/*前回見たk*/    if (pk != k) {        if (~pk)ina.close();        edges.clear();        pk = k;        name = (k == 6) ? "C:\\Users\\kaout\\Desktop\\undi_sizek\\roku.txt" : "C:\\Users\\kaout\\Desktop\\undi_sizek\\undi_size" + tos(k) + ".txt";        ina = ifstream(name);        rem = lims[k];        rep(i, k)rep(j, i + 1, k)edges.emplace_back(i, j);        pk = k;    }    undigraph<int> g(k);    if (rem == 0) {        g.resize(0);        return g;    }    int m;    ina >> m;    while (m) {        int lb = lbit(m);        int id = log2(lb);        g.add(edges[id].first, edges[id].second);        m ^= lb;    }    rem--;    return g;}
vector<tree<int>> trees(int k) {    vector<tree<int>> res;    while (1) {        tree<int> g = next_tree(k);        if (sz(g) == 0)break;        res.push_back(g);    }    return res;}
vector<undigraph<int>> undis(int k) {    vector<undigraph<int>> res;    while (1) {        undigraph<int> g = next_undi(k);        if (sz(g) == 0)break;        res.push_back(g);    }    return res;}
template<class T,class I> vector<vector<int>> table(graph<T> &g,I init_value) {    vvi(res, g.n, g.n,init_value);    rep(i, g.n) { forg_f(gi, g[i]) { forg_f_init(g[i]);res[i][t] = c; }}    return res;}

#ifdef type_id_graph
template<class T> vector<vector<edge_<T>>> type_list(digraph<T> &g) {    vector<vector<edge_<T>>> res;    rep(i, g.n) { forg_f(gi, g[i]) { forg_f_init(g[i]);res[ty].push_back(g[i][gi]); }}    return res;}
template<class T> vector<vector<edge_<T>>> type_list(undigraph<T> &g, int types = -1) {    int tn = types;    if (types == -1)tn = g.n;    rep(i, g.n) { forg_f(gi, g[i]) { forg_f_init(g[i]);chma(tn, ty); }}    vector<vector<edge_<T>>> res(tn + 1);    vi was(g.n);    rep(i, g.n) {        forg_f(gi, g[i]) {   forg_f_init(g[i]);         if (f < t)res[ty].push_back(g[i][gi]);            else if (f == t && !was[f]) {                res[ty].push_back(g[i][gi]);                was[f] = 1;            }        }    }    return res;}
//idは 00 11 22のようにedgesに持たれている
template<class T> vi krus_id(undigraph<T> &g) {    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    vi res;    fora(e, E) { if (uf.unite(e.f, e.t)) { res.push_back(e.id); }}    return res;}
//graの方が早い
#endif
//type,idが使いたい場合は
//type_id_graph
// を付ける
#define edge edge_
/*@formatter:on*/}

//幾何ライブラリ   *@formatter:off*/
bool eq(double a, double b){return fabs(a-b) < eps;}
vp ort_next_ = {{1, 0}, {0,  1}, {-1, 0}, {0,  -1}, {1,  0}};
vp ort_prev_ = {{1, 0}, {1,  0}, {0, 1}, {-1,  0}, {0,  -1}};
class D {
public:
    double x, y;
    D(): x(0), y(0) {}
    D(double x, double y): x(x), y(y) {}
    D(P p): x(p.first), y(p.second){}
    D& operator+=( D& v) { x += v.x; y += v.y; return *this;}
    D operator+( D& v)  { return D(*this) += v;}
    D& operator-=( D& v) { x -= v.x; y -= v.y; return *this;}
    D operator-( D& v)  { return D(*this) -= v;}
    D& operator*=(double s) { x *= s; y *= s; return *this;}
    D operator*(double s)  { return D(*this) *= s;}
    D& operator/=(double s) { x /= s; y /= s; return *this;}
    D operator/(double s)  { return D(*this) /= s;}
    bool operator==(D & v) {return x == v.x && y == v.y;}
    //内積
    double dot( D& v)  { return x*v.x + y*v.y;}
    //外積
    double cross( D& v)  { return x*v.y - v.x*y;}
    //距離の2乗
    double norm2()  { return x*x + y*y;}
    double norm()  { return sqrt(norm2());}
    //単位ベクトルを返す
    D unit(){return (*this)/ norm();}
    // 象限
    int ort()  {        if (abs(x) < eps && abs(y) <= eps) return 0;        if (y > 0) return x>0 ? 1 : 2;        else return x<0 ? 3 : 4;    }
    //偏角でソート出来る
    bool operator<( D& v)  {        int o = ort(), vo = v.ort();        if (o != vo) return o < vo;        return cross(v) > 0;    }
    bool operator>( D &v)  {        int o = ort(), vo = v.ort();        if (o != vo) return o > vo;        return cross(v) < 0;    }
    //反時計回りで隣接する象限との境界にある、長さ1の点を返す
    //(+,+)で呼ぶと(0, 1)が返る
    D ort_bound_next(){        int i=ort();        assert(i);        return D(ort_next_[i]);    }
    //時計回り
    D ort_bound_prev(){        int i=ort();        assert(i);        return D(ort_prev_[i]);    }
};
using dots = vector<D>;
double dis( D &p,  D &q);

bool eq(D a, D b){ return dis(a,b) <= eps;}

//線分
struct E{
    D f,t;
    E(D f=D(),D t=D()):f(f),t(t){}
};

//円
struct R{
    D o;//中心
    double r;//半径
    R():o(D(0,0)),r(0){}
    R(D o, dou r):o(o),r(r){}
    bool operator==(R b){return b.o == o && b.r == r;}
};

istream& operator>>(istream& ist, D& v) {ist >> v.x >> v.y; return ist;}
istream& operator>>(istream& ist, R& v) {ist >> v.o >> v.r; return ist;}
ostream& operator<<(ostream& os,  D& v) {os<<"("<<v.x<<","<<v.y<<")"; return os;}
ostream& operator<<(ostream& os,  R& v) {os<<"("<<v.o<<","<<v.r<<")"; return os;}
D vect(dou theta, dou len, D a = D());
double arg(D b, D a = D());
//@基本計算 取得***********************************************

//外積
double cross( D p,  D q) { return p.x * q.y - p.y * q.x; }
//距
double dis2( D &p,  D &q){return ((p.x-q.x)*(p.x-q.x)) +(p.y-q.y)*(p.y-q.y) ;}
double dis( D &p,  D &q){return std::sqrt(dis2(p,q)) ;}

//@判定*********************************************************
//線分の交差判定 端点を認めない
bool kousa_mid(D fa,D ta,D fb,D tb) {    dou ax = fa.x;    dou ay = fa.y;    dou bx = ta.x;    dou by = ta.y;    dou cx = fb.x;    dou cy = fb.y;    dou dx = tb.x;    dou dy = tb.y;  dou ta1 = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);  dou tb1 = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);  dou tc1 = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);  dou td1 = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);  return tc1 * td1 < 0 && ta1 * tb1 < 0;}
bool kousa_mid(E f,E t){return kousa_mid(f.f,f.t,t.f,t.t);}
//端点を認める
bool kousa_hasi(D fa,D ta,D fb,D tb) {    dou ax = fa.x;    dou ay = fa.y;    dou bx = ta.x;    dou by = ta.y;    dou cx = fb.x;    dou cy = fb.y;    dou dx = tb.x;    dou dy = tb.y;  dou ta1 = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);  dou tb1 = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);  dou tc1 = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);  dou td1 = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);   return tc1 * td1 <= 0 && ta1 * tb1 <= 0;}
bool kousa_hasi(E f,E t){return kousa_hasi(f.f,f.t,t.f,t.t);}
//交差する点を0~2個返す 同じ円が渡された場合、空を返す
dots kousa(R a, R b) {    if(a==b)return dots();auto d = dis(a.o, b.o);    if (d > a.r + b.r)return dots();    if (eq(d, a.r + b.r))return dots{(a.o + b.o) / 2};    dou u = acos(((a.r * a.r) + (d * d) - (b.r * b.r)) / (2 * a.r * d));    dou t = arg(b.o - a.o);    return dots{vect(t + u, a.r, a.o), vect(t - u, b.r, a.o)};}


//@角度系 中心とするものの引数は右に*****************************

//偏角 [-pi,pi] aを中心としたbの偏角
double arg(D b, D a /*= D()*/) {b -= a;    auto d= atan2(b.y, b.x);    return d;}

//aを中心にbを何シータ 反時計回りに回転させたらcになるかを返す
//[0, 2pi)
double ang(D b,D c,D a=D(0,0)){    c-=a;    b-=a;    int ob = b.ort();    int oc = c.ort();    if(!ob || !oc){        cerr<<"同じ点は渡せない"<<endl;        re();    }    double ret=0;    if(ob < oc){        ret += (oc-ob-1) * PI/2;        ret += arg(b.ort_bound_next()) - arg(b);        /*prevが-piではなくpiになる       */ if(oc==3){            ret += arg(c)-(-PI);        }else{            ret += arg(c)- arg(c.ort_bound_prev());        }    }else if(ob>oc){        return PI*2 - ang(c,b,D(0,0));    }else{        if(b > c){            return PI*2 - ang(c,b,D(0,0));        }else{            ret = arg(c) - arg(b);        }    }    return ret;}
//角度 aを中心とする最小角度の絶対値
double uang(D b, D c, D a = D(0, 0)) {            dou ret = ang(b, c, a);            return min(ret, 2 * PI - ret);        }
//@余弦定理*******************************************************
//todo

//@変形***********************************************************
//bを回転 aを中心に反時計回り
D rot(D b,dou theta,D a=D(0,0)){    b-=a;    dou x=b.x,y=b.y;    return D(cos(theta) * x -sin(theta) * y, sin(theta) * x + cos(theta)*y)+a;}
D mid(D& a,D& b){return (a+b)/2;}
D change_norm(D b, double want_norm, D a = D(0, 0)) {    b -= a;    b /= b.norm();    b *= want_norm;    return b + a;}
//偏角と長さから座標を返す[aを中心とする]
D vect(dou theta, dou len, D a /*= D()*/){return D(cos(theta) * len, sin(theta) * len) + a;}

//@アルゴリズム***************************************************
// 一直線上の3点を含めない
// 全ベクトルが同じ方向でどれを足し合わせるかを考えるときに一直線にすると爆発するhttps://atcoder.jp/contests/abc139/tasks/abc139_f F-Engines
//反時計回り
vector<D> convexhull(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) { return ps; }    vector<D> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](D &l, D &r) { if (l.x != r.x)return l.x < r.x; else return l.y < r.y; });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) <= eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) <= eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}
//インデックスを返す
vi convexhull_i(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};else return vi{0, 1};}vi res(2 * n);vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) <= eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) <= eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}
// 一直線上の3点を含める
vector<D> convexhull_line(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) { return ps; }    vector<D> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](D &l, D &r) { if (l.x != r.x)return l.x < r.x; else return l.y < r.y; });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}
vi convexhull_line_i(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};        else return vi{0, 1};    }    vi res(2 * n);    vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}/*@formatter:on*/
//D := dots
//R(D, len)
//dis, dis2(距離の2乗)
//kousa

//#ifndef epsdou
//a //epsdou
//#endif
//long doubleが重い場合は切る
//mapにDを入れるとバグる
void solve() {
    vector<R> S;
    din(sx, sy, tx, ty);
    S.emplace_back(D(sx, sy), 0);
    S.emplace_back(D(tx, ty), 0);
    in(N);
    rep(N) {
        din(x, y, r);
        S.emplace_back(D(x, y), r);
    }
    undigraph<double > g(1010);;
    N += 2;
    rep(i, N) {
        rep(j, N) {
            if (i == j)continue;
            dou cost = dis(S[i].o, S[j].o) - S[i].r - S[j].r;
            deb(i, j, cost);
            g.add(i, j, max(0, cost));
        }
    }
    auto dis = dijkstra(g, 0, -1);
    deb(dis);
    out(dis[1]);
}

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: 129244