//#undef _DEBUG
//#pragma GCC optimize("Ofast")
//不動小数点の計算高速化
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;

#ifndef _DEBUG
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
namespace cmp = boost::multiprecision;
using cint = cmp :: cpp_int;
using bdou = cmp::cpp_dec_float_100;
#endif
#define ll long long
using sig_dou = double;
//Tの方が大きい型
#define maxmin_lr(S, T)T max(S a, T b){return a > b ? a : b;}T max(T a, S b){return a > b ? a : b;}T min(S a, T b){return a < b ? a : b;}T min(T a, S b){return a < b ? a : b;}
//int < ll < bint < dou < ldou
maxmin_lr(int, ll)
maxmin_lr(int, __int128)
maxmin_lr(int, double)
maxmin_lr(int, long double)
maxmin_lr(ll, __int128)
maxmin_lr(ll, double)
maxmin_lr(ll, long double)
maxmin_lr(__int128, double)
maxmin_lr(__int128, long double)
maxmin_lr(double, long double)
//epsdou 下で


#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


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

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

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


#ifdef _DEBUG
string message;
string res_mes;
//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
//区間削除は出来ない
template<class T> struct my_pbds_tree {    set<T> s;    auto begin() { return s.begin(); }    auto end() { return s.end(); }    auto rbegin() { return s.rbegin(); }    auto rend() { return s.rend(); }    auto empty() { return s.empty(); }    auto size() { return s.size(); }    void clear() { s.clear(); }    template<class U> void insert(U v) { s.insert(v); }template<class U> void operator+=(U v) { insert(v); }    template<class F> auto erase(F v) { return s.erase(v); }    template<class U> auto find(U v) { return s.find(v); }    template<class U> auto lower_bound(U v) { return s.lower_bound(v); }    template<class U> auto upper_bound(U v) { return s.upper_bound(v); }    auto find_by_order(ll k) {        auto it = s.begin();        for (ll i = 0; i < k; i++)it++;        return it;    }    auto order_of_key(ll v) {        auto it = s.begin();        ll i=0;        for (;it != s.end() && *it <v ; i++)it++;        return i;    }};
#define pbds(T) my_pbds_tree<T>
//gp_hash_tableでcountを使えないようにするため
template<class T,class U> struct my_unordered_map{    unordered_map<T,U> m;    my_unordered_map(){};    auto begin(){        return m.begin();    }    auto end(){return m.end();}    auto cbegin(){return m.cbegin();}    auto cend(){return m.cend();}    template<class V>auto erase(V v){return m.erase(v);}    void clear(){m.clear();}    /*countは gp_hash_tableに存在しない*/    /*!= m.end()*/    template<class V>auto find(V v){return m.find(v);}    template<class V>auto & operator [](V n) { return m[n] ;}};
#define unordered_map my_unordered_map
#define umapi unordered_map<ll,ll>
#define umapp unordered_map<P,ll>
#define umapu unordered_map<uint64_t,ll>
#define umapip unordered_map<ll,P>
template<class T,class U, class X> auto count (unordered_map<T,U>& a, X k){return a.find(k) != a.end();}
#else
#define endl '\n'
//umapはunorderd_mapになる
//umapiはgp_hash_table

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

#define umapi __gnu_pbds::gp_hash_table<ll,ll,xorshift>
#define umapp __gnu_pbds::gp_hash_table<P,ll,xorshift>
#define umapu __gnu_pbds::gp_hash_table<uint64_t,ll,xorshift>
#define umapip __gnu_pbds::gp_hash_table<ll,P,xorshift>
template<class T,class U,class W, class X>auto count (__gnu_pbds::gp_hash_table<T,U,W>& a, X k){return a.find(k) != a.end();}
#endif


/*@formatter:on*/
struct xorshift {
    static uint64_t splitmix64(uint64_t x) {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
        return x ^ (x >> 31);
    }
    size_t operator()(uint64_t x) const {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }
    size_t operator()(std::pair<ll, ll> x) const {
        ll v = ((x.first) << 32) | x.second;
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(v + FIXED_RANDOM);
    }
};
/*@formatter:off*/
template<class U, class L> void operator+=(__gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update> &s, L v) { s.insert(v); }
//衝突対策
#define ws ws_

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

template<class T>T MAX(){return numeric_limits<T>::max();}
template<class T>T MIN(){return numeric_limits<T>::min();}


//@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;
const double eps = 1e-9;
//基本コメントアウト
//struct epsdou {    double v;    epsdou(double v = 0) : v(v) {}    epsdou(ll v = 0) : v(v) {}    epsdou(signed v = 0) : v(v) {}    template<class T> epsdou &operator+=(T b) {        v += (double) b;        return (*this);    }    template<class T> epsdou &operator-=(T b) {        v -= (double) b;        return (*this);    }    template<class T> epsdou &operator*=(T b) {        v *= (double) b;        return (*this);    }    template<class T> epsdou &operator/=(T b) {        v /= (double) b;        return (*this);    }    epsdou operator+(epsdou b) { return v + (double) b; }    epsdou operator-(epsdou b) { return v - (double) b; }    epsdou operator*(epsdou b) { return v * (double) b; }    epsdou operator/(epsdou b) { return v / (double) b; }    epsdou operator-() const { return epsdou(-v); }    template<class T> bool operator<(T b) { return v < (double) b; }    template<class T> bool operator>(T b) { return v > (double) b; }    template<class T> bool operator==(T b) { return fabs(v - (double) b) <= eps; }    template<class T> bool operator<=(T b) { return v < (double) b || fabs(v - b) <= eps; }    template<class T> bool operator>=(T b) { return v > (double) b || fabs(v - b) <= eps; }    operator double() { return v; }};istream &operator>>(istream &iss, epsdou &a) {    iss >> a.v;    return iss;}ostream &operator<<(ostream &os, epsdou &a) {    os << a.v;    return os;}
//#define eps_conr_t(o) template<class T> epsdou operator o(T 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(/)
//epsdou max(signed a, epsdou b) { return (epsdou)a > b ? (epsdou)a : b; }
//epsdou max(epsdou a, signed b) { return a > (epsdou)b ? a : (epsdou)b; }
//epsdou min(signed a, epsdou b) { return (epsdou)a < b ? (epsdou)a : b; }
//epsdou min(epsdou a, signed b) { return a < (epsdou)b ? a : (epsdou)b; }
//#undef double
//#define double epsdou
/*@formatter:on*/
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<>epsdou MIN(){return numeric_limits<T>::min();}

#undef double
#define double epsdou
#undef dou
#define dou epsdou

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

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

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

#define PQ priority_queue<ll, vector<ll>, greater<ll> >
#define tos to_string
using mapi = map<ll, ll>;
using mapp = map<P, ll>;
using mapd = map<dou, ll>;
using mapc = map<char, ll>;
using maps = map<str, ll>;
using seti = set<ll>;
using setp = set<P>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<ll>;
#define uset unordered_set
#define useti unordered_set<ll,xorshift>
#define mset multiset
#define mseti multiset<ll>
#define umap unordered_map
#define mmap multimap
//任意のマクロサポート用 使う度に初期化する
int index_, v1_, v2_, v3_;

template<class T> struct pq {
    priority_queue<T, vector<T>, greater<T> > q;/*小さい順*/    T su = 0;
    void clear() {
        q = priority_queue<T, vector<T>, greater<T> >();
        su = 0;
    }
    void operator+=(T v) {
        su += v;
        q.push(v);
    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {
        su -= q.top();
        q.pop();
    }
    T poll() {
        T ret = q.top();
        su -= ret;
        q.pop();
        return ret;
    }
    ll size() { return q.size(); }
};
template<class T> struct pqg {
    priority_queue<T> q;/*大きい順*/    T su = 0;
    void clear() {
        q = priority_queue<T>();
        su = 0;
    }
    void operator+=(T v) {
        su += v;
        q.push(v);
    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {
        su -= q.top();
        q.pop();
    }
    T poll() {
        T ret = q.top();
        su -= ret;
        q.pop();
        return ret;
    }
    ll size() { return q.size(); }
};
#define pqi pq<ll>
#define pqgi pqg<ll>
//マクロ 繰り返し
//↓@オーバーロード隔離

#define rep1(n) for(ll rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)
#define rep2(i, n) for(ll i = 0,rep2lim=n; i < rep2lim ; ++i)
#define rep3(i, m, n) for(ll i = m,rep3lim=n; i < rep3lim ; ++i)
#define rep4(i, m, n, ad) for(ll i = m,rep4lim=n; i < rep4lim ; i+= ad)
//逆順 閉区間
#define rer2(i, n) for(ll i = n; i >= 0 ; i--)
#define rer3(i, m, n) for(ll i = m,rer3lim=n; i >= rer3lim ; i--)
#define rer4(i, m, n, dec) for(ll i = m,rer4lim=n; i >= rer4lim ; i-=dec)
//ループを一つにまとめないとフォーマットで汚くなるため
#define nex_ind1(i) i++
#define nex_ind2(i, j, J) i = (j + 1 == J) ? i + 1 : i, j = (j + 1 == J ? 0 : j + 1)
#define nex_ind3(i, j, k, J, K)i = (j + 1 == J && k + 1 == K) ? i + 1 : i, j = (k + 1 == K) ? (j + 1 == J ? 0 : j + 1) : j, k = (k + 1 == K ? 0 : k + 1)
#define nex_ind4(i, j, k, l, J, K, L) i = (j + 1 == J && k + 1 == K && l + 1 == L) ? i + 1 : i, j = (k + 1 == K && l + 1 == L) ? (j + 1 == J ? 0 : j + 1) : j, k = (l + 1 == L ?(k + 1 == K ? 0 : k + 1) : k), l = l + 1 == L ? 0 : l + 1
#define nex_ind5(i, j, k, l, m, J, K, L, M) i = (j + 1 == J && k + 1 == K && l + 1 == L && m + 1 == M) ? i + 1 : i, j = (k + 1 == K && l + 1 == L && m + 1 == M) ? (j + 1 == J ? 0 : j + 1) : j, k = (l + 1 == L && m + 1 == M ?(k + 1 == K ? 0 : k + 1) : k), l =  m + 1 == M ? l+1 == L ? 0 : l+1 : l,  m = m + 1 == M ? 0 : m + 1

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

//i,j,k...をnまで見る
#define reps2(i, n) repss2(i, n)
#define reps3(i, j, n) repss4(i, j, n, n)
#define reps4(i, j, k, n) repss6(i, j, k, n, n, n)
#define reps5(i, j, k, l, n) repss8(i, j, k, l, n, n, n, n)

template<class T> void nex_repv2(int &i, int &j, int &I, int &J, vector<vector<T>> &s) {
    while (1) {
        j++;
        if (j >= J) {
            j = 0;
            i++;
            if (i < I) { J = (int) s[i].size(); }
        }
        if (i >= I || J) return;
    }
}
template<class T> void nex_repv3(int &i, int &j, int &k, int &I, int &J, int &K, vector<vector<vector<T>>> &s) {
    while (1) {
        k++;
        if (k >= K) {
            k = 0;
            j++;
            if (j >= J) {
                j = 0;
                i++;
                if (i >= I)return;
            }
        }
        J = (int) s[i].size();
        K = (int) s[i][j].size();
        if (J && K) return;
    }
}

#define repv_2(i, a) repss2(i, sz(a))
//正方形である必要はない
//直前を持つのとどっちが早いか
#define repv_3(i, j, a) for (int 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]))

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

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

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

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

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

template<class U> vector<U> to1d(vector<U> &a) { return a; }
template<class U> vector<U> to1d(vector<vector<U>> &a) {
    vector<U> res;
    for (auto &&a1 : a)for (auto &&a2 : a1)res.push_back(a2);
    return res;
}
template<class U> vector<U> to1d(vector<vector<vector<U>>> &a) {
    vector<U> res;
    for (auto &&a1 : a)for (auto &&a2 : a1)for (auto &&a3 : a2) res.push_back(a3);
    return res;
}
template<class U> vector<U> to1d(vector<vector<vector<vector<U>>>> &a) {
    vector<U> res;
    for (auto &&a1 : a)for (auto &&a2 : a1)for (auto &&a3 : a2) for (auto &&a4 : a3)res.push_back(a4);
    return res;
}
template<class U> vector<U> to1d(vector<vector<vector<vector<vector<U>>>>> &a) {
    vector<U> res;
    for (auto &&a1 : a)for (auto &&a2 : a1)for (auto &&a3 : a2) for (auto &&a4 : a3)for (auto &&a5 : a4)res.push_back(a5);
    return res;
}
template<class U> vector<U> to1d(vector<vector<vector<vector<vector<vector<U>>>>>> &a) {
    vector<U> res;
    for (auto &&a1 : a)for (auto &&a2 : a1)for (auto &&a3 : a2) for (auto &&a4 : a3)for (auto &&a5 : a4)for (auto &&a6 : a5)res.push_back(a6);
    return res;
}

#define fora_init_2(a, A) ;
#define fora_init_3(fora_i, a, A) auto &&a = A[fora_i];
#define fora_init_4(a, b, A, B) auto &&a = A[fora_i]; auto &&b = B[fora_i];
#define fora_init_5(fora_i, a, b, A, B) auto &&a = A[fora_i]; auto &&b = B[fora_i];
#define fora_init_6(a, b, c, A, B, C) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i];
#define fora_init_7(fora_i, a, b, c, A, B, C) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i];
#define fora_init_8(a, b, c, d, A, B, C, D) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i]; auto && d = D[fora_i];
#define fora_init_9(fora_i, a, b, c, d, A, B, C, D) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i]; auto && d = D[fora_i];
#define fora_init(...) over9(__VA_ARGS__,fora_init_9, fora_init_8, fora_init_7, fora_init_6, fora_init_5, fora_init_4, fora_init_3, fora_init_2)(__VA_ARGS__)

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


#define forp_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 form_init(k, v, ...) auto &&k = (*form_it).fi;auto &&v = (*form_it).se;
#define forrm_init(k, v, ...) auto &&k = (*forrm_it).fi;auto &&v = (*forrm_it).se;
#define fors_init(v, ...) auto &&v = (*fors_it);

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

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

#define forr_2(a, A) rer(forr_i, sz(A)-1)
#define forr_3(forr_i, a, A) rer(forr_i, sz(A)-1)
#define forr_4(a, b, A, B) rer(forr_i, sz(A)-1)
#define forr_5(forr_i, a, b, A, B) rer(forr_i, sz(A)-1)
#define forr_6(a, b, c, A, B, C) rer(forr_i, sz(A)-1)
#define forr_7(forr_i, a, b, c, A, B, C) rer(forr_i, sz(A)-1)
#define forr_8(a, b, c, d, A, B, C, D) rer(forr_i, sz(A)-1)
#define forr_9(forr_i, a, b, c, d, A, B, C, D) rer(forr_i, sz(A)-1)

//↑@オーバーロード隔離

//rep系はインデックス、for系は中身
#define rep(...) over4(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)
#define rer(...) over4(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)
//char用のrep
#define repc(i, m, n) for(char i = m,repc3lim=n; i < repc3lim ; ++i)
//i,j,k...をnまで見る
#define reps(...) over5(__VA_ARGS__,reps5,reps4,reps3,reps2,)(__VA_ARGS__)
#define repss(...) over10(__VA_ARGS__, repss10, a, repss8, a, repss6, a, repss4, a, repss2) (__VA_ARGS__)
//vectorのindexを走査する
//repv(i,j,vvi)
#define repv(...) over6(__VA_ARGS__,repv_6,repv_5,repv_4,repv_3,repv_2,)(__VA_ARGS__)
#define rerv(i, A) for (int i = sz(A)-1; i >= 0 ; i--)
//repvn(dp) nは次元
#define repv1(a) repv(i, a)
#define repv2(a) repv(i, j, a)
#define repv3(a) repv(i, j, k, a)
#define repv4(a) repv(i, j, k, l, a)

#define fora(...) over9(__VA_ARGS__, fora_9, fora_8, fora_7, fora_6, fora_5, fora_4, fora_3, fora_2)(__VA_ARGS__)
#define forr(...) over9(__VA_ARGS__, forr_9, forr_8, forr_7, forr_6, forr_5, forr_4, forr_3, forr_2)(__VA_ARGS__)

//#define forr(v, a) for(auto&& v : makeReversedRange(a))
//参照を取らない
#define forv(a, b) for(auto a : to1d(b))

//インデックスを前後含めて走査
#define ring(i, s, len) for (int i = s, prev = (s == 0) ? len - 1 : s - 1, next = (s == len - 1) ? 0 : s + 1, cou = 0; cou < len; cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
//値と前後を見る
#define ringv(v, d) index_=0;for (auto prev = d[sz(d)-1],next= (int)d.size()>1?d[1]:d[0],v = d[0]; index_ < sz(d); index_++, prev = v, v = next, next = (index_>=sz(d)-1?d[0]:d[index_+1]))
// 左右をnext prevで見る 0の左と nの右
#define forlr(v, d, banpei_l, banpei_r) rep(forlr_i,sz(d))

#define form(...) over5(__VA_ARGS__,form5,form4,form3,form2,form1)(__VA_ARGS__)
#define forrm(...) over5(__VA_ARGS__,forrm5,forrm4,forrm3,forrm2,forrm1)(__VA_ARGS__)
#define fors(...) over4(__VA_ARGS__,fors4,fors3,fors2,fors1)(__VA_ARGS__)
#define forslr(...) over6(__VA_ARGS__,forslr6,forslr5,forslr4,forslr3)(__VA_ARGS__)
#define 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__)


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

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

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

//埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
    template<> class hash<std::pair<signed, signed>> { public:size_t operator()(const std::pair<signed, signed> &x) const { return hash<ll>()(((ll) x.first << 32) | x.second); }};
    template<> class hash<std::pair<ll, ll>> { public:/*大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断*/size_t operator()(const std::pair<ll, ll> &x) const { return hash<ll>()(((ll) x.first << 32) | x.second); }};
}
//stream まとめ
istream &operator>>(istream &iss, P &a) {
    iss >> a.first >> a.second;
    return iss;
}
template<typename T> istream &operator>>(istream &iss, vector<T> &vec_) {
    for (T &x: vec_) iss >> x;
    return iss;
}
template<class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
    os << p.fi << " " << p.se;
    return os;
}
ostream &operator<<(ostream &os, T p) {
    os << p.f << " " << p.s << " " << p.t;
    return os;
}
ostream &operator<<(ostream &os, F p) {
    os << p.a << " " << p.b << " " << p.c << " " << p.d;
    return os;
}
template<typename T> ostream &operator<<(ostream &os, vector<T> &vec_) {
    for (ll i = 0; i < vec_.size(); ++i)os << vec_[i] << (i + 1 == vec_.size() ? "" : " ");
    return os;
}
template<typename T> ostream &operator<<(ostream &os, vector<vector<T>> &vec_) {
    for (ll i = 0; i < vec_.size(); ++i) {
        for (ll j = 0; j < vec_[i].size(); ++j) { os << vec_[i][j] << " "; }
        os << endl;
    }
    return os;
}
template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {
    os << endl;
    for (auto &&v:m) os << v << endl;
    return os;
}
template<class T> ostream &operator<<(ostream &os, set<T> s) {
    fora(v, s) { os << v << " "; }
    return os;
}
template<class T> ostream &operator<<(ostream &os, mset<T> s) {
    fora(v, s) { os << v << " "; }
    return os;
}
template<class T> ostream &operator<<(ostream &os, deque<T> a) {
    fora(v, a)os << v << " ";
    return os;
}
ostream &operator<<(ostream &os, vector<vector<char>> &vec_) {
    rep(h, sz(vec_)) {
        rep(w, sz(vec_[0])) { os << vec_[h][w]; }
        os << endl;
    }
    return os;
}
//template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>& a) {fora(v,a)os<<v<<endl;return os;}
template<typename W, typename H> void resize(W &vec_, const H head) { vec_.resize(head); }
template<typename W, typename H, typename ... T> void resize(W &vec_, const H &head, const T ... tail) {
    vec_.resize(head);
    for (auto &v: vec_)resize(v, tail...);
}
//template<typename W, typename H> void resize(vector<W> &vec, const H head) { vec.resize(head); }
//template<typename W, typename H, typename ... T> void resize(vector<W> &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);}
template<typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool all_of2(vector<T> &v, F f) {
    rep(i, sz(v)) { if (!all_of2(v[i], f))return false; }
    return true;
}
template<typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool any_of2(vector<T> &v, F f) {
    rep(i, sz(v)) { if (any_of2(v[i], f))return true; }
    return false;
}
template<typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool none_of2(vector<T> &v, F f) {
    rep(i, sz(v)) { if (none_of2(v[i], f))return false; }
    return true;
}
template<typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll find_if2(vector<T> &v, F f) {
    rep(i, sz(v)) { if (find_if2(v[i], f))return i; }
    return sz(v);
}
template<typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll rfind_if2(vector<T> &v, F f) {
    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }
    return -1;
}
template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }
template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }
template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }
template<typename T, typename F> ll count_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll count_if2(vector<T> &vec_, F f) {
    ll ret = 0;
    fora(v, vec_)ret += count_if2(v, f);
    return ret;
}
template<typename T, typename F> void for_each2(T &v, F f) { f(v); }
template<typename T, typename F> void for_each2(vector<T> &vec_, F f) { fora(v, vec_)for_each2(v, f); }
template<typename W> ll count_od(vector<W> &a) { return count_if2(a, [](ll v) { return v & 1; }); }
template<typename W> ll count_ev(vector<W> &a) { return count_if2(a, [](ll v) { return !(v & 1); }); }
//削除した後のvectorを返す
template<typename T, typename F> vector<T> erase_if2(vector<T> &v, F f) {
    vector<T> nv;
    rep(i, sz(v)) { if (!f(v[i])) { nv.push_back(v[i]); }}
    return nv;
}
template<typename T, typename F> vector<vector<T>> erase_if2(vector<vector<T>> &v, F f) {
    vector<vector<T>> res;
    rep(i, sz(v)) { res[i] = erase_if2(v[i], f); }
    return res;
}
template<typename T, typename F> vector<T> l_erase_if2(vector<T> &v, F f) {
    vector<T> nv;
    rep(i, sz(v)) { if (!f(v[i])) { nv.push_back(v[i]); }}
    return nv;
}

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

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

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

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

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

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

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

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

#define find_if__2(a, right) find_if2(a,lamr(right))
#define find_if__3(a, v, siki) find_if2(a,[&](auto v){return siki;})
#define find_if(...) over3(__VA_ARGS__,find_if__3,find_if__2)(__VA_ARGS__)
#define find_if_f(a, f) find_if2(a,f)

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

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

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

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

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

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

//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);

#define for_eached__2(a, right) [&](auto a) {auto b = a;for_each(b, right);return b; }(a)
#define for_eached__3(a, v, shori) [&](auto a) {auto b = a;for_each(b, v, shori);return b; }(a)
#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 erase_if_s__2(a, right) l_erase_if2(a,lamr(right))
#define erase_if_s__3(a, v, siki) l_erase_if2(a,[&](auto v){return siki;})
#define erase_if_s(...) over3(__VA_ARGS__,erase_if_s__3,erase_if_s__2)(__VA_ARGS__)


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

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


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


template<class T, class U, class W> void replace(vector<W> &a, T key, U v) { rep(i, sz(a))if (a[i] == key)a[i] = v; }
template<class T, class U, class W> void replace(vector<vector<W>> &A, T key, U v) { rep(i, sz(A))replace(A[i], key, v); }
void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }
void replace(str &a, char key, char v) { replace(all(a), key, v); }
//keyと同じかどうか01で置き換える
template<class T, class U> void replace(vector<T> &a, U k) { rep(i, sz(a)) a[i] = a[i] == k; }
template<class T, class U> void replace(vector<vector<T >> &a, U k) { rep(i, sz(a))rep(j, sz(a[0])) a[i][j] = a[i][j] == k; }
//template<class T> void replace(T &a) { replace(a, '#'); }
void replace(str &a) {
    int dec = 0;
    if ('a' <= a[0] && a[0] <= 'z')dec = 'a';
    if ('A' <= a[0] && a[0] <= 'Z')dec = 'A';
    fora(v, a)v -= dec;
}
void replace(str &a, str key, str v) {
    stringstream t;
    ll kn = sz(key);
    std::string::size_type Pos(a.find(key));
    ll l = 0;
    while (Pos != std::string::npos) {
        t << a.substr(l, Pos - l);
        t << v;
        l = Pos + kn;
        Pos = a.find(key, Pos + kn);
    }
    t << a.substr(l, sz(a) - l);
    a = t.str();
}
template<class T> bool includes(vector<T> &a, vector<T> &b) {
    vi c = a;
    vi d = b;
    sort(all(c));
    sort(all(d));
    return includes(all(c), all(d));
}
template<class T> bool is_permutation(vector<T> &a, vector<T> &b) { return is_permutation(all(a), all(b)); }
template<class T> bool next_permutation(vector<T> &a) { return next_permutation(all(a)); }
void iota(vector<ll> &ve, ll s, ll n) {
    ve.resize(n);
    iota(all(ve), s);
}
vi iota(ll s, ll len) {
    vi ve(len);
    iota(all(ve), s);
    return ve;
}
template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {
    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;
    rep(i, sz(a))res.eb(a[i], b[i]);
    return res;
}
template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {
    a.resize(sz(p)), b.resize(sz(p));
    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;
}
template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {
    assert(sz(a) == sz(b) && sz(b) == sz(c));
    vector<T2<A, B, C>> res;
    rep(i, sz(a))res.eb(a[i], b[i], c[i]);
    return res;
}
template<class A, class B, class C, class D> auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));
    vector<F2<A, B, C, D>> res;
    rep(i, sz(a))res.eb(a[i], b[i], c[i], d[i]);
    return res;
}
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {
    fisiti,
    fisitd,
    fisdti,
    fisdtd,
    fdsiti,
    fdsitd,
    fdsdti,
    fdsdtd,
    fitisi,
    fitisd,
    fitdsi,
    fitdsd,
    fdtisi,
    fdtisd,
    fdtdsi,
    fdtdsd,
    sifiti,
    sifitd,
    sifdti,
    sifdtd,
    sdfiti,
    sdfitd,
    sdfdti,
    sdfdtd,
    sitifi,
    sitifd,
    sitdfi,
    sitdfd,
    sdtifi,
    sdtifd,
    sdtdfi,
    sdfdfd,
    tifisi,
    tifisd,
    tifdsi,
    tifdsd,
    tdfisi,
    tdfisd,
    tdfdsi,
    tdfdsd,
    tisifi,
    tisifd,
    tisdfi,
    tisdfd,
    tdsifi,
    tdsifd,
    tdsdfi,
    tdsdfd
};
template<class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {
    typedef pair<A, B> U;
    if (type == fisi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
};
template<class U> void sort(vector<U> &a, pcomparator type) { if (type == fisi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; }); };
template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D> > &a, pcomparator type) {
    typedef F2<A, B, C, D> U;
    if (type == fisi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });
};
template<class U> void sort(vector<U> &a, tcomparator type) {
    if (type == 0) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });
    else if (type == 1) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });
    else if (type == 2) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });
    else if (type == 3) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });
    else if (type == 4) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });
    else if (type == 5) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });
    else if (type == 6) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });
    else if (type == 7) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });
    else if (type == 8) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });
    else if (type == 9) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });
    else if (type == 10) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });
    else if (type == 11) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });
    else if (type == 12) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });
    else if (type == 13) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });
    else if (type == 14) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });
    else if (type == 15) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });
    else if (type == 16) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });
    else if (type == 17) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });
    else if (type == 18) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });
    else if (type == 19) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });
    else if (type == 20) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });
    else if (type == 21) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });
    else if (type == 22) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });
    else if (type == 23) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });
    else if (type == 24) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });
    else if (type == 25) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });
    else if (type == 26) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });
    else if (type == 27) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });
    else if (type == 28) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });
    else if (type == 29) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });
    else if (type == 30) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });
    else if (type == 31) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });
    else if (type == 32) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });
    else if (type == 33) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });
    else if (type == 34) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });
    else if (type == 35) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });
    else if (type == 36) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });
    else if (type == 37) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });
    else if (type == 38) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });
    else if (type == 39) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });
    else if (type == 40) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == 41) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == 42) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == 43) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f; }); else if (type == 44) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == 45) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == 46) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == 47) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });
}
template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {
    typedef F2<A, B, C, D> U;
    if (type == 0) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });
    else if (type == 1) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });
    else if (type == 2) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });
    else if (type == 3) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });
    else if (type == 4) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });
    else if (type == 5) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });
    else if (type == 6) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });
    else if (type == 7) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });
    else if (type == 8) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });
    else if (type == 9) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });
    else if (type == 10) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });
    else if (type == 11) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });
    else if (type == 12) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });
    else if (type == 13) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });
    else if (type == 14) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });
    else if (type == 15) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });
    else if (type == 16) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });
    else if (type == 17) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });
    else if (type == 18) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });
    else if (type == 19) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });
    else if (type == 20) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });
    else if (type == 21) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });
    else if (type == 22) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });
    else if (type == 23) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });
    else if (type == 24) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });
    else if (type == 25) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });
    else if (type == 26) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });
    else if (type == 27) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });
    else if (type == 28) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });
    else if (type == 29) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });
    else if (type == 30) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });
    else if (type == 31) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });
    else if (type == 32) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });
    else if (type == 33) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });
    else if (type == 34) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });
    else if (type == 35) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });
    else if (type == 36) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });
    else if (type == 37) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });
    else if (type == 38) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });
    else if (type == 39) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });
    else if (type == 40) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == 41) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == 42) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == 43) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a; }); else if (type == 44) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == 45) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == 46) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == 47) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });
}

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

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

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

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


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

#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__)


#ifdef _DEBUG
template<class T> void err2(T &&head) { cerr << head; }
template<class T, class... U> void err2(T &&head, U &&... tail) {
    cerr << head << " ";
    err2(tail...);
}
template<class T, class... U> void err(T &&head, U &&... tail) {
    cerr << head << " ";
    err2(tail...);
    cerr << "" << endl;
}
template<class T> void err(T &&head) { cerr << head << endl; }
void err() { cerr << "" << endl; }
template<class T> string out_m2(vector<T> &a, ll W = inf) {
    stringstream ss;
    if (W == inf)W = min(sz(a), 20ll);
    if (sz(a) == 0)return ss.str();
    rep(i, W) {
        ss << a[i];
        if (typeid(a[i]) == typeid(P)) { ss << endl; } else { ss << " "; }
    }
    return ss.str();
}
template<class T> string out_m2(vector<vector<T> > &a, vi H, vi W, int key = -1) {
    stringstream ss;
    ss << endl;
    vi lens(sz(W));
    fora(h, H) {
        rep(wi, sz(W)) {
            int v = a[h][W[wi]];
            str s = tos(v);
            if (abs(v) == inf || abs(v) == linf)s = "e";
            lens[wi] = max(lens[wi], sz(s) + 1);
            lens[wi] = max(lens[wi], sz(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) {
            str s = tos(a[h][w]);
            if (abs(a[h][w]) == inf || abs(a[h][w]) == linf)s = "e";
            ss << std::right << std::setw(lens[wi]) << s;
            wi++;
        }
        ss << "" << endl;
    }
    return ss.str();
}
template<class T> string out_m2(vector<vector<T> > &a, ll H = inf, ll W = inf, int key = -1) {
    H = (H != inf) ? H : min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    vi hs, ws;
    rep(h, H) { hs.push_back(h); }
    rep(w, W) { ws.push_back(w); }
    return out_m2(a, hs, ws, key);
}
template<class T> string out_m2(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 out_m2(b, H, W, key);
}
template<class T> string out_m2(vector<vector<vector<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {
    stringstream ss;
    if (H == inf)H = 12;
    H = min(H, sz(a));
    rep(i, H) {
        ss << endl;
        ss << out_m2(a[i], W, U, i);
    }
    return ss.str();
}
template<class T, size_t N> string out_m2(T (&a)[N]) {
    vector<T> b;
    resize(b, N);
    rep(i, N) { b[i] = a[i]; }
    return out_m2(b);
}
template<class T, size_t N, size_t M> string out_m2(T (&a)[N][M]) {
    vector<vector<T>> b;
    resize(b, N, M);
    rep(i, N) { rep(j, M) { b[i][j] = a[i][j]; }}
    return out_m2(b);
}
template<class T, size_t N, size_t M, size_t O> string out_m2(T (&a)[N][M][O]) {
    vector<vector<vector<T>>> b;
    resize(b, N, M, O);
    rep(i, N) { rep(j, M) { rep(k, O) { b[i][j][k] = a[i][j][k]; }}}
    return out_m2(b);
}
string out_m2(int a) {
    stringstream ss;
    ss << a;
    return ss.str();
}
template<class T> string out_m2(mvec<mvec<T> > &a, ll H = inf, ll W = inf, int key = inf) {
    H = min({H, sz(a), 6ll});
    W = min({W, sz(a[0]), 6ll});
    stringstream ss;
    ss << endl;
    if (key == inf)ss << " *|";
    else {
        ss << std::right << std::setw(2) << key;
        ss << "|";
    }
    rep(w, -W, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 8 + 3)ss << "_";
    ss << "" << endl;
    rep(h, -H, H) {
        ss << std::right << std::setw(2) << h << "|";
        int NW = min(sz(a[h]), 6ll);
        rep(w, -NW, NW) { if (abs(a[h][w]) == linf) ss << "   e" << ""; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    return ss.str();
}
template<class T> string out_m2(mvec<mvec<mvec<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {
    stringstream ss;
    if (H == inf)H = 6;
    H = min(H, sz(a));
    rep(i, -H, H) {
        ss << endl;
        ss << out_m2(a[i], W, U, i);
    }
    return ss.str();
}
template<class T> string out_m2(T &a) {
    stringstream ss;
    ss << a;
    return ss.str();
}
template<class T> string out_m(vector<T> &a, ll W = inf) {
    stringstream ss;
    if (W == inf)W = min(sz(a), 12ll);
    if (sz(a) == 0)return ss.str();
    rep(i, W) { ss << a[i] << " "; }
    ss << "" << endl;
    return ss.str();
}
template<class T> string out_m(vector<vector<T> > &a, ll H = inf, ll W = inf, int key = -1) {
    H = min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    stringstream ss;
    ss << endl;
    if (key == -1)ss << " *|"; else ss << " " << key << "|";
    rep(w, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 4 + 3)ss << "_";
    ss << "" << endl;
    rep(h, H) {
        ss << std::right << std::setw(2) << h << "|";
        rep(w, min(sz(a[h]), 12ll)) { if (abs(a[h][w]) == linf) ss << "   e" << ""; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    ss << endl;
    return ss.str();
}
template<class T> string out_m(vector<vector<vector<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {
    stringstream ss;
    if (H == inf)H = 5;
    H = min(H, sz(a));
    rep(i, H) {
        ss << endl;
        ss << out_m(a[i], W, U, i);
    }
    ss << endl;
    return ss.str();
}
string out_m(int a) {
    stringstream ss;
    ss << a << endl;
    return ss.str();
}
template<class T> string out_m(T &a) {
    stringstream ss;
    ss << a << endl;
    return ss.str();
}
template<class T> void outv(vector<T> &a, ll W = inf) { cout << out_m(a, W) << endl; }
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf, int key = -1) { cout << out_m(a, H, W, key) << endl; }
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf, ll U = linf) { cout << out_m(a, H, W, U) << endl; }

template<class T> void out2(T head) {
    cout << head;
    res_mes += out_m2(head);
}
template<class T, class... U> void out2(T head, U ... tail) {
    cout << head << " ";
    res_mes += out_m2(head) + " ";
    out2(tail...);
}
template<class T, class... U> void out(T head, U ... tail) {
    cout << head << " ";
    res_mes += out_m2(head) + " ";
    out2(tail...);
    cout << "" << endl;
    res_mes += "\n";
}
template<class T> void out(T head) {
    cout << head << endl;
    res_mes += out_m2(head) + "\n";
}
void out() { cout << "" << endl; }

#else
template<class T> void outv(vector<T> &a, ll W = inf) {
    rep(i, min(W, sz(a))) { cout << a[i] << " "; }
    cout << "" << endl;
}
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf, int key = -1) { rep(i, min(H, sz(a))) { outv(a[i], W); }}
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf, ll U = linf) { ; }
#define err(...);

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(vector<T> &a, int n = inf) { rep(i, min(n, sz(a)))cout << a[i] << endl; }
//テーブルをスペースなしで出力
template<class T> void outt(vector<vector<T>> &a) {
    rep(i, sz(a)) {
        rep(j, sz(a[i])) { cout << a[i][j]; }
        cout << endl;
    }
}
//int型をbit表記で出力
void outb(int a) { cout << bitset<20>(a) << endl; }
template<class T> void na(vector<T> &a, ll n) {
    a.resize(n);
    rep(i, n)cin >> a[i];
}
template<class T> void na(set<T> &a, ll n) { rep(i, n)a.insert(in()); }
#define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai];
#define dnad(a, n) vi a(n); rep(dnai,n) cin >> a[dnai],a[dnai]--;
template<class T> void nao(vector<T> &a, ll n) {
    a.resize(n + 1);
    a[0] = 0;
    rep(i, n)cin >> a[i + 1];
}
template<class T> void naod(vector<T> &a, ll n) {
    a.resize(n + 1);
    a[0] = 0;
    rep(i, n)cin >> a[i + 1], a[i + 1]--;
}
template<class T> void nad(vector<T> &a, ll n) {
    a.resize(n);
    rep(i, n)cin >> a[i], a[i]--;
}
template<class T> void nad(set<T> &a, ll n) { rep(i, n)a.insert(in() - 1); }
template<class T, class U> void na2(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n);
    b.resize(n);
    rep(i, n)cin >> a[i] >> b[i];
}
template<class T, class U> void na2(set<T> &a, set<U> &b, ll n) {
    rep(i, n) {
        a.insert(in());
        b.insert(in());
    }
}
#define dna2(a, b, n) vi a(n),b(n);rep(dna2i, n)cin >> a[dna2i] >> b[dna2i];
template<class T, class U> void nao2(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n + 1);
    b.resize(n + 1);
    a[0] = b[0] = 0;
    rep(i, n)cin >> a[i + 1] >> b[i + 1];
}
#define dna2d(a, b, n) vi a(n),b(n);rep(dna2di, n){cin >> a[dna2di] >> b[dna2di];a[dna2di]--,b[dna2di]--;}
template<class T, class U> void na2d(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n);
    b.resize(n);
    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;
}
template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i];
}
#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(dna3i, n)cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
template<class T, class U, class W> void na3d(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;
}
#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  rep(dna3di, n){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}
template<class T, class U, class W, class X> void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    d.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i] >> d[i];
}
#define dna4(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i];
#define dna4d(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i],--a[dna4i] ,-- b[dna4i],-- c[dna4i],--d[dna4i];
#define nt(a, h, w) resize(a,h,w);rep(nthi,h)rep(ntwi,w) cin >> a[nthi][ntwi];
#define ntd(a, h, w) resize(a,h,w);rep(ntdhi,h)rep(ntdwi,w) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(ntphi,1,h+1)rep(ntpwi,1,w+1) cin >> a[ntphi][ntpwi];
#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 << " " <<

#define deb1(x)  debugName(x)<<" = "<<out_m2(x)
#define deb_2(x, ...) deb1(x) <<", "<< deb1(__VA_ARGS__)
#define deb_3(x, ...) deb1(x) <<", "<< deb_2(__VA_ARGS__)
#define deb_4(x, ...) deb1(x) <<", "<< deb_3(__VA_ARGS__)
#define deb5(x, ...) deb1(x) <<", "<< deb_4(__VA_ARGS__)
#define deb6(x, ...) deb1(x) <<", "<< deb5(__VA_ARGS__)
#define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)
#define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)
#define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)
#define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)


#ifdef _DEBUG
bool was_deb = false;
#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)

#define base_keta 8
void print_n_base(int x, int base) { cerr << bitset<base_keta>(x) << endl; }
template<class T> void print_n_base(vector<T> X, int base) {
    cerr << endl;
    for (auto &&x:X) { print_n_base(x, base); }
    cerr << endl;
}

//n進数
#define deb2(x) 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)<<" = "<<out_m2(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 = out_m2(ex_v, sz(ex_v));debugName(v);cerr<<" = "<<endl;cerr << S << endl;} while (0);
template<class T, class F> string out_m2_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 out_m2(a, hs, ws_, key);
}
template<class T, class F> string out_m2_f(vector<vector<vector<T>>> &a, F f) {
    stringstream ss;
    int H = sz(a);
    if (sz(a) == 0)return ss.str();
    rep(i, H) { ss << out_m2_f(a[i], f, i); }
    ss << "" << endl;
    return ss.str();
}
#define debf_normal(tab, f) do{cerr<<debugName(tab)<<" = "<<endl;cerr<< out_m2_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(...) ;
#endif


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


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


using u32 = unsigned;
using u64 = unsigned long long;
using u128 = __uint128_t;

using bint =__int128;
std::ostream &operator<<(std::ostream &dest, __int128_t value) {    std::ostream::sentry s(dest);    if (s) {        __uint128_t tmp = value < 0 ? -value : value;        char buffer[128];        char *d = std::end(buffer);        do {            --d;            *d = "0123456789"[tmp % 10];            tmp /= 10;        } while (tmp != 0);        if (value < 0) {            --d;            *d = '-';        }        ll len = std::end(buffer) - d;        if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); }    }    return dest;}
__int128 to_bint(string &s) {    __int128 ret = 0;    for (ll i = 0; i < s.length(); ++i)        if ('0' <= s[i] && s[i] <= '9')            ret = 10 * ret + s[i] - '0';    return ret;}
void operator>>(istream &iss, bint &v) {
    string S;
    iss>>S;
    v = 0;
    rep(i,sz(S)){
        v*=10;
        v +=  S[i]-'0';
    }
}
//エラー
void ole() {
#ifdef _DEBUG
    debugline("ole");    exit(0);
#endif
    string a = "a";    rep(i, 30)a += a;    rep(i, 1 << 17)cout << a << endl;    cout << "OLE 出力長制限超過" << endl;    exit(0);}
void re() {    assert(0 == 1);    exit(0);}
void tle() { while (inf)cout << inf << endl; }

//便利関数

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

void ranvlr(ll n, ll min, ll max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    rep(i, n) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}
template<class T>vector<pair<T,int>> run_length(vector<T> &a) {vector<pair<T,int>> ret;ret.eb(a[0], 1);rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}return ret;}
vector<pair<char, ll>> run_length(string &a) {    vector<pair<char, ll>> ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}    return ret;}
template<class T, class F> T mgr(T ok, T ng, F f, int deb_ = 0) {    bool han = true;    if (deb_) {        if (ok < ng)            while (ng - ok > 1) {                T mid = (ok + ng) >> 1;                if (f(mid))ok = mid, han = true; else ng = mid, han = false;                deb(mid, han);            }        else            while (ok - ng > 1) {                T mid = (ok + ng) >> 1;                if (f(mid))ok = mid, han = true; else ng = mid, han = false;                deb(mid, han);            }    }else{        if (ok < ng)            while (ng - ok > 1) {                T mid = (ok + ng) >> 1;                if (f(mid))ok = mid, han = true; else ng = mid, han = false;            }        else            while (ok - ng > 1) {                T mid = (ok + ng) >> 1;                if (f(mid))ok = mid, han = true; else ng = mid, han = false;            }    }    return ok;}
template<class T, class F> T mgr(signed ok, T ng, F f) {    return mgr((T)ok, ng, f);}
template<class T, class F> T mgr(T ok, signed ng, F f) {    return mgr(ok, (T)ng, f);}
template<class F> int mgr(signed ok, signed ng, F f) {    return mgr((ll)ok, (ll)ng, f);}
//[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(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> dou mgrd(dou ok, dou ng, F f,int kai=100) {
    bool han = true;
    if (ok < ng)
        rep(i,kai) {
            dou mid = (ok + ng) /2;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            deb(mid,han);
        }
    else
        rep(i,kai) {
            dou mid = (ok + ng) /2;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            deb(mid,han);
        }
    return ok;
}
//strを整数として比較
string smax(str &a, str b) {    if (sz(a) < sz(b)) { return b; }    else if (sz(a) > sz(b)) { return a; }    else if (a < b)return b;    else return a;}
//strを整数として比較
string smin(str &a, str b) {    if (sz(a) > sz(b)) { return b; }    else if (sz(a) < sz(b)) { return a; }    else if (a > b)return b;    else return a;}
//エラー-1
template<typename W, typename T> ll find(vector<W> &a, int l, const T key) {rep(i, l, sz(a))if (a[i] == key)return i;return -1;}
template<typename W, typename T> ll find(vector<W> &a, const T key) {    rep(i, sz(a))if (a[i] == key)return i;    return -1;}
template<typename W, typename T> P find(vector<vector<W >> &a, const T key) {    rep(i, sz(a))rep(j, sz(a[0]))if (a[i][j] == key)return mp(i, j);    return mp(-1, -1);}
//getid(find())を返す 1次元にする
template<typename W, typename T> int findi(vector<vector<W >> &a, const T key) {    rep(i, sz(a))rep(j, sz(a[0]))if (a[i][j] == key)return i*sz(a[0])+j;    return -1;}
template<typename W, typename U> T find(vector<vector<vector<W >>> &a, const U key) {    rep(i, sz(a))rep(j, sz(a[0]))rep(k, sz(a[0][0]))if (a[i][j][k] == key)return mt(i, j, k);    return mt(-1, -1, -1);}
//stringも書く
int find(string& s, const string key){    int klen = sz(key);    rep(i,sz(s)-klen+1){        if(s[i]!=key[0])continue;        if(s.substr(i,klen) == key){            return i;        }    }    return -1;}
int find(string& s, int l, const string key){    int klen = sz(key);    rep(i, l, sz(s)-klen+1){        if(s[i]!=key[0])continue;        if(s.substr(i,klen) == key){            return i;        }    }    return -1;}
int find(string& s, const char key){    rep(i,sz(s)){        if(s[i]==key)return i;    }    return -1;}
int find(string& s, int l, const char key){    rep(i, l, sz(s)){        if(s[i]==key)return i;    }    return -1;}
template<typename W, typename T> ll count2(W &a, const T k) { return a == k; }
template<typename W, typename T> ll count2(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
template<typename W, typename T> ll count(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
vi count(vi &a) {    int ma = 0;    fora(v, a) {        if (ma < v)ma = v;    }    vi res(ma + 1);    fora(v, a) {        res[v]++;    }    return res;}
ll count(str &a, str k) {    ll ret = 0, len = k.length();    auto pos = a.find(k);    while (pos != string::npos)pos = a.find(k, pos + len), ++ret;    return ret;}
//'a' = 'A' = 0 として集計 既に-'a'されていても動く
vi count(str &a, int l, int r) {    vi cou(26);    char c = 'a';    if ('A' <= a[l] && a[l] <= 'Z')c = 'A';    if ('a' <= a[l] && a[l] <= 'z') c = 'a'; else c = 0;    rep(i, l, r)++cou[a[i] - c];    return cou;}
vi count(str &a, int r = inf){return count(a, 0, min(r, sz(a)));}
#define couif count_if
//algorythm
ll rev(ll a) {    ll res = 0;    while (a) {        res *= 10;        res += a % 10;        a /= 10;    }    return res;}
template<class T> auto rev(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(string &a) { string b = a; reverse(all(b));return b; }
constexpr ll p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};

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

ll mask10(ll v) { return p10[v] - 1; }
//変換系
//[v] := iとなるようなvectorを返す
//存在しない物は-1
template<class T> auto keys(T a) {    vector<decltype((a.begin())->fi)> res;    for (auto &&k :a)res.push_back(k.fi);    return res;}
template<class T> auto values(T a) {    vector<decltype((a.begin())->se)> res;    for (auto &&k :a)res.push_back(k.se);    return res;}
template<class T, class U>  bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}
template<class T, class U>  bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}
template<class T> constexpr  T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }
template<class T> constexpr  T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }
template<class T>  T min(vector<T>& a) { return *min_element(all(a)); }
template<class T>  T mini(vector<T>& a) { return min_element(all(a)) - a.begin(); }
template<class T>  T min(vector<T>& a, ll n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T min(vector<T>& a, ll s, ll n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a) { return *max_element(all(a)); }
template<class T,class U>  T max(vector<T>& a,vector<U>& b) { return max(*max_element(all(a)),*max_element(all(b))); }
template<class T>  T maxi(vector<T>& a) { return max_element(all(a)) - a.begin(); }
template<class T>  T max(vector<T>& a, ll n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a, ll s, ll n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<typename A, size_t N> A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O> A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}
template<typename A, size_t N> A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O> A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}
template<class T> T sum(vector<T> &v, ll s, ll t) {    T ret = 0;    rep(i, s, min(sz(v), t))ret += v[i];    return ret;}
template<class T> T sum(vector<T> &v, ll t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<T> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<T> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<T>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<T>> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<vector<T>>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<vector<T>>> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<vector<vector<T>>>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<vector<vector<T>>>> > &v, int t = inf) { return sum(v, 0, t); }

template<class T> auto sum(priority_queue<T, vector<T>, greater<T> > &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}
template<class T> auto sum(priority_queue<T> &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}
template<class T> T mul(vector<T> &v, ll t = inf) {    T ret = v[0];    rep(i, 1, min(t, sz(v)))ret *= v[i];    return ret;}

//template<class T, class U, class... W>  auto sumn(vector<T> &v, U head, W... tail) {    auto ret = sum(v[0], tail...);    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);    return ret;}
vi v_i(vi &a) {    int n = max(a) + 1;    vi ret(n, -1);    rep(i, sz(a)) {        ret[a[i]] = i;    }    return ret;}
void clear(PQ &q) { q = PQ(); }
void clear(priority_queue<int>&q) { q = priority_queue<int>();}
template<class T> void clear(queue<T> &q) { while (q.size())q.pop(); }
template<class T> T *negarr(ll size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}
/*@formatter:on*/
template<class T> T *negarr2(ll h, ll w) {
    double **dummy1 = new double *[2 * h + 1];
    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];
    dummy1[0] = dummy2 + w;
    for (ll i = 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }
    double **a = dummy1 + h;
    return a;
}
//imoは0-indexed
//ruiは1-indexed
template<class T> vector<T> imo(vector<T> &v) {
    vector<T> ret = v;
    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];
    return ret;
}
//kと同じものの数
template<class T, class U> vi imo(vector<T> &a, U k) {
    vector<T> ret = a;
    rep(i, sz(ret))ret[i] = a[i] == k;
    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];
    return ret;
}
template<class T> vector<T> imox(vector<T> &v) {
    vector<T> ret = v;
    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];
    return ret;
}
//漸化的に最小を持つ
template<class T> vector<T> imi(vector<T> &v) {
    vector<T> ret = v;
    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);
    return ret;
}
template<class T> vector<T> ima(vector<T> &v) {
    vector<T> ret = v;
    rep(i, sz(ret) - 1)chma(ret[i + 1], ret[i]);
    return ret;
}
template<class T> vector<T> rimi(vector<T> &v) {
    vector<T> ret = v;
    rer(i, sz(ret) - 1, 1)chmi(ret[i - 1], ret[i]);
    return ret;
}
template<class T> vector<T> rima(vector<T> &v) {
    vector<T> ret = v;
    rer(i, sz(ret) - 1, 1)chma(ret[i - 1], ret[i]);
    return ret;
}
template<class T> struct ruiC {
    vector<T> rui;
    ruiC(vector<T> &ru) : rui(ru) {}
    /*先頭0*/
    ruiC() : rui(1, 0) {}
    T operator()(ll l, ll r) {
        if (l > r) {
            cerr << "ruic ";
            deb(l, r);
            assert(0);
        }
        return rui[r] - rui[l];
    }
    T operator()(int r = 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> struct ruimax {
    template<typename Monoid> struct SegmentTree {        /*pairで処理*/        int sz;
        vector<Monoid> seg;
        const Monoid M1 = mp(MIN<T>(), -1);
        Monoid f(Monoid a, Monoid b) { return max(a, b); }
        void build(vector<T> &a) {
            int n = sz(a);
            sz = 1;
            while (sz < n) sz <<= 1;
            seg.assign(2 * sz, M1);
            rep(i, n) { seg[i + sz] = mp(a[i], i); }
            for (int k = sz - 1; k > 0; k--) { seg[k] = f(seg[k << 1], seg[(k << 1) | 1]); }
        }
        Monoid query(int a, int b) {
            Monoid L = M1, R = M1;
            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
                if (a & 1) L = f(L, seg[a++]);
                if (b & 1) R = f(seg[--b], R);
            }
            return f(L, R);
        }
        Monoid operator[](const int &k) const { return seg[k + sz]; }
    };
private:
    vector<T> ve;
    SegmentTree<pair<T, int>> seg;
    vector<T> rv;
    vector<T> ri;
    bool build = false;
public:
    int n;
    ruimax(vector<T> &a) : ve(a), n(sz(a)) {
        int index = -1;
        int ma = MIN<T>();
        rv.resize(n + 1);
        ri.resize(n + 1);
        rv[0] = -linf;
        ri[0] = -1;
        rep(i, n) {
            if (chma(ma, a[i])) { index = i; }
            rv[i + 1] = ma;
            ri[i + 1] = index;
        }
    }
    T operator()(int l, int r) {
        if (!(l <= r && 0 <= l && r <= n)) {
            deb(l, r, n);
            assert(0);
        }
        if (l == 0) { return rv[r]; }
        else {
            if (!build)seg.build(ve), build = true;
            return seg.query(l, r).first;
        }
    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    T getv(int l, int r) { return operator()(l, r); }
    T geti(int l, int r) {
        assert(l <= r && 0 <= l && r <= n);
        if (l == 0) { return ri[r]; }
        else {
            if (!build)seg.build(ve), build = true;
            return seg.query(l, r).second;
        }
    }
    T geti(int r = inf) { return geti(0, min(r, n)); };
    T getv(int r = inf) { return getv(0, min(r, n)); };
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};
template<class T> struct ruimin {
    template<typename Monoid> struct SegmentTree {  /*pairで処理*/         int sz;
        vector<Monoid> seg;
        const Monoid M1 = mp(MAX<T>(), -1);
        Monoid f(Monoid a, Monoid b) { return min(a, b); }
        void build(vector<T> &a) {
            int n = sz(a);
            sz = 1;
            while (sz < n) sz <<= 1;
            seg.assign(2 * sz, M1);
            rep(i, n) { seg[i + sz] = mp(a[i], i); }
            for (int k = sz - 1; k > 0; k--) { seg[k] = f(seg[k << 1], seg[(k << 1) | 1]); }
        }
        Monoid query(int a, int b) {
            Monoid L = M1, R = M1;
            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {
                if (a & 1) L = f(L, seg[a++]);
                if (b & 1) R = f(seg[--b], R);
            }
            return f(L, R);
        }
        Monoid operator[](const int &k) const { return seg[k + sz]; }
    };
private:
    vector<T> ve;
    SegmentTree<pair<T, int>> seg;
    vector<T> rv;
    vector<T> ri;
    bool build = false;
    int n;
public:
    ruimin(vector<T> &a) : ve(a), n(sz(a)) {
        int index = -1;
        int mi = MAX<T>();
        rv.resize(n + 1);
        ri.resize(n + 1);
        rv[0] = linf;
        ri[0] = -1;
        rep(i, n) {
            if (chmi(mi, a[i])) { index = i; }
            rv[i + 1] = mi;
            ri[i + 1] = index;
        }
    }
    T operator()(int l, int r) {
        assert(l <= r && 0 <= l && r <= n);
        if (l == 0) { return rv[r]; }
        else {
            if (!build)seg.build(ve), build = true;
            return seg.query(l, r).first;
        }
    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    T getv(int l, int r) { return operator()(l, r); }
    T geti(int l, int r) {
        assert(l <= r && 0 <= l && r <= n);
        if (l == 0) { return ri[r]; }
        else {
            if (!build)seg.build(ve), build = true;
            return seg.query(l, r).second;
        }
    }
    T geti(int r = inf) { return geti(0, min(r, n)); };
    T getv(int r = inf) { return getv(0, min(r, n)); };
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};

template<class T> ostream &operator<<(ostream &os, ruiC<T> a) {
    fora(v, a.rui)os << v << " ";
    return os;
}
template<class T> vector<T> ruiv(vector<T> &a) {
    vector<T> ret(a.size() + 1);
    rep(i, a.size())ret[i + 1] = ret[i] + a[i];
    return ret;
}
template<class T> ruiC<T> ruic() { return ruiC<T>(); }
template<class T> ruiC<T> ruic(vector<T> &a) {
    vector<T> ret = ruiv(a);
    return ruiC<T>(ret);
}
vvi() ruib(vi &a) {
    vvi(res, 61, sz(a) + 1);
    rep(k, 61) { rep(i, sz(a)) { res[k][i + 1] = res[k][i] + ((a[i] >> k) & 1); }}
    return res;
}
vector<ruiC<int>> ruibc(vi &a) {
    vector<ruiC<int>> ret(61);
    vvi(res, 61, sz(a));
    rep(k, 61) {
        rep(i, sz(a)) { res[k][i] = (a[i] >> k) & 1; }
        ret[k] = ruic(res[k]);
    }
    return ret;
}
vector<ll> ruiv(string &a) {
    if (sz(a) == 0)return vi(1);
    ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;
    vector<ll> ret(a.size() + 1);
    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;
    return ret;
}
ruiC<ll> ruic(string &a) {
    vector<ll> ret = ruiv(a);
    return ruiC<ll>(ret);
}
//kと同じものの数
template<class T, class U> vi ruiv(T &a, U k) {
    vi ret(a.size() + 1);
    rep(i, a.size())ret[i + 1] = ret[i] + (a[i] == k);
    return ret;
}
template<class T, class U> ruiC<ll> ruic(T &a, U k) {
    vi ret = ruiv(a, k);
    return ruiC<ll>(ret);
}
template<class T> struct ruiC2 {
    int H;
    vector<ruiC<T>> rui;
    ruiC2(vector<vector<T>> &ru) : rui(sz(ru)), H(sz(ru)) { for (int h = 0; h < H; h++) { rui[h] = ruic(ru[h]); }}
    //WについてHを返す
    vector<T> operator()(ll l, ll r) {
        if (l > r) {
            cerr << "ruic ";
            deb(l, r);
            assert(0);
        }
        vector<T> res(H);
        for (int h = 0; h < H; h++)res[h] = rui[h](l, r);
        return res;
    }
    //HについてWを返す
    ruiC<T> &operator[](ll h) {
        assert(h < H);
        return rui[h];
    }
    //    vector<T> operator()(int r) { return operator()(0, r); }
    /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
    /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
//    T operator[](ll i) { return rui[i + 1] - rui[i]; }
    /*0から順に追加される必要がある*/
//    T back() { return rui.back(); }
//    ll size() { return rui.size(); }
//    auto begin(){return rui.begin();}
//    auto end(){return rui.end();}
};

//a~zを0~25として
// rui(l,r)でvector(26文字について, l~rのcの個数)
// rui[h] ruic()を返す
ruiC2<ll> ruicou(str &a) {
    str s = a;
    replace(s);
    vector<ruiC<ll>> res(26);
    vvi(cou, 26, sz(s));
    rep(i, sz(s)) { cou[s[i]][i] = 1; }
    return ruiC2<ll>(cou);
}
ruiC2<ll> ruicou(vi &a) {
    int H = max(a) + 1;
    vector<ruiC<ll>> res(H);
    vvi(cou, H, sz(a));
    rep(i, sz(a)) { cou[a[i]][i] = 1; }
    return ruiC2<ll>(cou);
}
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);
}
//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;
}

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

//掛け算
template<class T> struct ruimulC {
    vector<T> rv;
    int n;
    ruimulC(vector<T> &a) : rv(a), n(sz(a)) {
        rv.resize(n + 1);
        rv[0] = 1;
        rep(i, n) { rv[i + 1] = a[i] * rv[i]; }
    }
    ruimulC() : n(0) {
        rv.resize(n + 1);
        rv[0] = 1;
    }
    void operator+=(T v) {
        rv.push_back(rv.back() * v);
        n++;
    }
    T operator()(int l, int r) {
        assert(l <= r && 0 <= l && r <= n);
        return rv[r] / rv[l];
    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};
template<class T> ruimulC<T> ruimul(vector<T> &a) { return ruimulC<T>(a); }
template<class T> ruimulC<T> ruimul() {
    vector<T> a;
    return ruimulC<T>(a);
}
/*@formatter:off*/
template<class T> T *rruim(vector<T> &a) {    ll len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    res[len - 1] = 1;    rer(i, len - 1)res[i - 1] = res[i] * a[i];    return res;}
template<class T, class U> void inc(pair<T,U> &a, U v = 1) { a.first += v,a.second+=v; }
template<class T, class U> void inc(T &a, U v = 1) { a += v; }
template<class T, class U = int> void inc(vector<T> &a, U v = 1) { for (auto &u:a)inc(u, v); }
template<class T, class U> void dec(T &a, U v = 1) { a -= v; }
template<class T, class U = int> void dec(vector<T> &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class U> void dec(string &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class T,class U, class W> void dec(vector<T> &a,vector<U> &b, W v = 1) { for (auto &u :a)dec(u, v);for (auto &u :b)dec(u, v); }
template<class T,class U,class W> void dec(vector<T> &a,vector<U> &b,vector<W>&c ) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1);for (auto &u :c)dec(u, 1); }
bool ins(ll h, ll w, ll H, ll W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool 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;}
bool d_(int a, int b) {if (b == 0)return false;return (a % b) == 0;}

//添え字を返す
template<class F> ll goldd_l(ll left, ll right, F calc) {    double GRATIO = 1.6180339887498948482045868343656;    ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));    ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));    ll fl = calc(lm);    ll fr = calc(rm);    while (right - left > 10) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + (ll) ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    ll minScore = MAX<ll>();    ll resIndex = left;    for (ll i = left; i < right + 1; ++i) {        ll score = calc(i);        if (minScore > score) {            minScore = score;            resIndex = i;        }    }    return resIndex;}
template<class F> ll goldt_l(ll left, ll right, F calc) {        double GRATIO = 1.6180339887498948482045868343656;        ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));        ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));        ll fl = calc(lm);        ll fr = calc(rm);        while (right - left > 10) {            if (fl > fr) {                right = rm;                rm = lm;                fr = fl;                lm = left + (ll) ((right - left) / (GRATIO + 1.0));                fl = calc(lm);            } else {                left = lm;                lm = rm;                fl = fr;                rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));                fr = calc(rm);            }        }    if (left > right) {        ll l = left;        left = right;        right = l;    }    ll maxScore = MIN<ll>();    ll resIndex = left;    for (ll i = left; i < right + 1; ++i) {        ll score = calc(i);        if (maxScore < score) {            maxScore = score;            resIndex = i;        }    }    return resIndex;}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template<class F> dou goldd_d(dou left, dou right, F calc, ll loop = 200) {    dou GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    /*200にすればおそらく大丈夫*/    /*余裕なら300に*/    ll k = 141;    loop++;    while (--loop) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
template<class F> dou goldt_d(dou left, dou right, F calc, ll loop = 200) {    double GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    loop++;    while (--loop) {        if (fl > fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
//l ~ rを複数の区間に分割し、極致を与えるiを返す time-20 msまで探索
template<class F> ll goldd_ls(ll l, ll r, F calc, ll time = 2000) {    auto lim = milliseconds(time - 20);    ll mini = 0, minv = MAX<ll>();    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        ll haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        ll nl = l;        ll nr = l + haba;        rep(i, k) {            ll ni = goldd_l(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> ll goldt_ls(ll l, ll r, F calc, ll time = 2000) {    auto lim = milliseconds(time - 20);    ll maxi = 0, maxv = MIN<ll>();    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        ll haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        ll nl = l;        ll nr = l + haba;        rep(i, k) {            ll ni = goldt_l(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class F> dou goldd_d_s(dou l, dou r, F calc, ll time = 2000) {    /*20ms余裕を持つ*/    auto lim = milliseconds(time - 20);    dou mini = 0, minv = MAX<dou>();    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldd_d(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> dou goldt_d_s(dou l, dou r, F calc, ll time = 2000) {    /*20ms余裕を残している*/    auto lim = milliseconds(time - 20);    dou maxi = 0, maxv = MIN<dou>();    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldt_d(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class T> T min(vector<vector<T >> &a) {    T res = MAX<T>();    rep(i, a.size())chmi(res, *min_element(all(a[i])));    return res;}
template<class T> T max(vector<vector<T >> &a) {    T res = MIN<T>();    rep(i, a.size())chma(res, *max_element(all(a[i])));    return res;}
template<class T> T min(vector<vector<vector<T>>> &a) {    T res = MAX<T>();    rep(i, a.size())chmi(res, min(a[i]));    return res;}
template<class T> T max(vector<vector<vector<T>>> &a) {    T res = MIN<T>();    rep(i, a.size())chma(res, max(a[i]));    return res;}
template<class T> T min(vector<vector<vector<vector<T>>>> &a) {    T res = MAX<T>();    rep(i, a.size())chmi(res, min(a[i]));    return res;}
template<class T> T max(vector<vector<vector<vector<T>>>> &a) {    T res = MIN<T>();    rep(i, a.size())chma(res, max(a[i]));    return res;}
template<class T> T min(vector<vector<vector<vector<vector<T>>>>> &a) {    T res = MAX<T>();    rep(i, a.size())chmi(res, min(a[i]));    return res;}
template<class T> T max(vector<vector<vector<vector<vector<T>>>>> &a) {    T res = MIN<T>();    rep(i, a.size())chma(res, max(a[i]));    return res;}
template<class T> T min(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {    T res = MAX<T>();    rep(i, a.size())chmi(res, min(a[i]));    return res;}
template<class T> T max(vector<vector<vector<vector<vector<vector<T>>>>>> &a) {    T res = MIN<T>();    rep(i, a.size())chma(res, max(a[i]));    return res;}
//pow周りの仕様
//powiを使うと整数型
//powbを使うとbint型
//powを使うと powlに変換され long doubleが返る
#ifdef _DEBUG
//整数値の場合はpowiを使った方がいいというメッセージを出すための物
auto my_powl(ll a, ll k){return powl(a,k);}
auto pow(ll a,ll k){    static bool was = 1;    if(was){message +="if integer use *powi* it's very fast\n";}    was=0;    return my_powl(a,k);}
//上のメッセージを出すための関数
auto pow(signed a,ll k){    return pow((ll)a,k);}
auto pow(signed a,signed k){    return pow((ll)a,(ll)k);}
#endif

//整数型のpow
int powi(int a, int k) {    if (a == 2)return 1ll << k;    int res = 1;    int x = a;    while (k) {        if(k & 1)res *= x;        x *= x;        k >>= 1;    }    return res;}
//define pow powlより上に動かすとバグる
bint pow(bint a, ll k) {    bint res = 1;    bint x = a;    while (k) {        if(k & 1)res *= x;        x *= x;        k >>= 1;    }    return res;}
bint pow(bint a, signed k) {return pow(a,(ll)k);}
bint powb(int a, int b){return pow((bint)a,b);}
constexpr bool bget(ll m, ll keta) {
    #ifdef _DEBUG
        assert(keta<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (m >> keta) & 1; }
//bget(n)次元
//                          NならN-1まで
vector<vi> bget2(vi& a,int keta_size){    vvi(res,keta_size,sz(a));    rep(k,keta_size){        rep(i, sz(a)){            res[k][i]=bget(a[i],k);        }    }    return res;}
vi bget1(vi& a,int keta){    vi res(sz(a));    rep(i, sz(a)){        res[i]=bget(a[i],keta);    }    return res;}
ll bget(ll m, ll keta, ll sinsuu) {    m /= (ll) pow(sinsuu, keta);    return m % sinsuu;}
constexpr ll bit(ll n) {
    #ifdef _DEBUG
        assert(n<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (1LL << (n)); }
ll bit(ll n, ll sinsuu) { return (ll) pow(sinsuu, n); }
ll mask(ll n) { return (1ll << n) - 1; }
//aをbitに置きなおす
//{0, 2} -> 101
ll bit(vi&a){    int m=0;    for(auto&& v:a) m|=bit(v);    return m;}

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


//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;}
//[i] := vを返す
//aは0~n-1で置き換えられる
vi compress(vi &a) {    vi b;    ll len = a.size();    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, umap<ll, ll> &map) {    vi b;    ll len = a.size();    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) {        ll v = a[i];        a[i] = lower_bound(all(b), a[i]) - b.begin();        map[v] = a[i];    }    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r) {    vi b;    ll len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r, vi &s) {    vi b;    ll len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    fora(v, s)b.push_back(v);    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    for (ll i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vi> &a) {    vi b;    fora(vv, a)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vector<vi >> &a) {    vi b;    fora(vvv, a)fora(vv, vvv)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
void compress(ll a[], ll len) {    vi b;    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }}
//要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(all(a),v))
#define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin())
#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())
#define rlowerIndex(a, v) (upper_bound(all(a),v)-a.begin()-1)
#define rupperIndex(a, v) (lower_bound(all(a),v)-a.begin()-1)

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

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

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

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

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

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

ll gcd(ll a, ll b) {while (b) a %= b, swap(a, b);return abs(a);}
ll gcd(vi b) {ll res = b[0];rep(i, 1, sz(b))res = gcd(b[i], res);return res;}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll lcm(vi a) {ll res = a[0];rep(i, 1, sz(a))res = lcm(a[i], res);return res;}
ll ceil(ll a, ll b) {if (b == 0) {debugline("ceil");deb(a, b);ole();return -1;} else if (a < 0) { return 0; } else { return (a + b - 1) / b; }}

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

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

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

//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;}

template<class T> vector<T> sub(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(vector<T> &A, int r) { return sub(A, 0, r); }
template<class T> vector<T> subn(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);}

constexpr int bsetlen = k5 * 2;
//constexpr int bsetlen = 5050;
#define bset bitset<bsetlen>
bool operator<(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] < b[i])return true;if (a[i] > b[i])return false;}return false;}
bool operator>(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] > b[i])return true;if (a[i] < b[i])return false;}return false;}
bool operator<=(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] < b[i])return true;if (a[i] > b[i])return false;}return true;}
bool operator>=(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] > b[i])return true;if (a[i] < b[i])return false;}return true;}
string operator~(string &a) {string res = a;for (auto &&c:res) {if (c == '0')c = '1';else if (c == '1')c = '0';else {cerr << "cant ~" << a << "must bit" << endl;exit(0);}}return res;}
ostream &operator<<(ostream &os, bset& a) {    bitset<10> b;    vi list;    rep(i,bsetlen){        if(a[i])list.push_back(i),b[i]=1;    }    os<<b<<", "<<list;    return os;}
int hbiti(bset&a){rer(i,bsetlen){if(a[i])return i;}return -1;}
#define hk(a, b, c) (a <= b && b < c)

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

template<class T> 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, class U> void operator+=(pair<T,U> &a, pair<T,U> & b) {a.fi+=b.fi;a.se+=b.se;}
template<class T, class U> pair<T,U> operator+(pair<T,U> &a, pair<T,U> & b) {return pair<T,U>(a.fi+b.fi,a.se+b.se);}
template<typename CharT, typename Traits, typename Alloc> basic_string<CharT, Traits, Alloc> operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rv) {
#ifdef _DEBUG
    static bool was = false;
    if (!was)message += "str += 65  is  'A'  not \"65\" ";
    was = true;
#endif
    return lhs + (char) rv;
}
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+(vector<T> &a, U v) {    vector<T> ret = a;    ret += v;    return ret;}template<class T, class U> vector<T> operator+(U v, vector<T> &a) {    vector<T> ret = a;    ret.insert(ret.begin(), v);    return ret;}template<class T> vector<T> operator+(vector<T> a, vector<T> b) {    vector<T> ret;    ret = a;    fora(v, b)ret += v;    return ret;}template<class T> vector<T> &operator+=(vector<T> &a, vector<T> &b) {    rep(i, sz(b)) {/*こうしないとa+=aで両辺が増え続けてバグる*/        a.push_back(b[i]); }    return a;}template<class T, class U> map<T, U> &operator+=(map<T, U> &a, map<T, U> &b) {    fora(bv, b) { a[bv.first] += bv.second; }    return a;}
template<class T> 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(vector<T> &v, unsigned ll i) { v.erase(v.begin() + i); }
template<typename T> void erase(vector<T> &v, unsigned ll s, unsigned ll e) { v.erase(v.begin() + s, v.begin() + e); }
template<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); }
template<typename T> void insert(set<T> &v, vector<T> list) { for (auto &&va :list)v.insert(va); }
template<class T> T poll(set<T>& S){T ret = *S.begin();S.erase(S.begin());return ret;}
template<class T> T poll(mset<T>& S){T ret = *S.begin();S.erase(S.begin());return ret;}
template<class T> T poll_back(set<T>& S){T ret = *S.rbegin();S.erase(S.rbegin());return ret;}
template<class T> T poll_back(mset<T>& S){T ret = *S.rbegin();S.erase(S.rbegin());return ret;}
template<class T> T peek(set<T>& S){T ret = *S.begin();return ret;}
template<class T> T peek(mset<T>& S){T ret = *S.begin();return ret;}
template<class T> T peek_back(set<T>& S){T ret = *S.rbegin();return ret;}
template<class T> T peek_back(mset<T>& S){T ret = *S.rbegin();return ret;}
vector<string> split(const string a, const char deli) {    string b = a + deli;    ll l = 0, r = 0, n = b.size();    vector<string> res;    rep(i, n) {        if (b[i] == deli) {            r = i;            if (l < r)res.push_back(b.substr(l, r - l));            l = i + 1;        }    }    return res;}
vector<string> split(const string a, const string deli) {    vector<string> res;    ll kn = sz(deli);    std::string::size_type Pos(a.find(deli));    ll l = 0;    while (Pos != std::string::npos) {        if (Pos - l)res.push_back(a.substr(l, Pos - l));        l = Pos + kn;        Pos = a.find(deli, Pos + kn);    }    if (sz(a) - l)res.push_back(a.substr(l, sz(a) - l));    return res;}
ll stoi(string& s){return stol(s);}
#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 T> vector<vector<T>> cut(vector<T> &a, int tou = 2) {    int N = sz(a);    vector<vector<T>> res(tou);    int hab = N / tou;    vi lens(tou, hab);    rep(i, N % tou) { lens[tou - 1 - i]++; }    int l = 0;    rep(i, tou) {        int len = lens[i];        int r = l + len;        res[i].resize(len);        std::copy(a.begin() + l, a.begin() + r, res[i].begin());        l = r;    }    return res;}
//長さn毎に分割する
template<typename T> vector<vector<T>> cutn(vector<T> &a, int len) {    int N = sz(a);    vector<vector<T>> res(ceil(N, len));    vi lens(N / len, len);    if (N % len)lens.push_back(N % len);    int l = 0;    rep(i, sz(lens)) {        int len = lens[i];        int r = l + len;        res[i].resize(len);        std::copy(a.begin() + l, a.begin() + r, res[i].begin());        l = r;    }    return res;}
vi inds_(vi &a) {    int n = sz(a);    vb was(n);    vi res(n);    rep(i, n) {        assert(!was[a[i]]);        res[a[i]] = i;        was[a[i]] = true;    }    return res;}
//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
#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

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;
}
//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);
}
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 &range(int l, int r) {static vi ret;ret.resize(r - l);rep(v, l, r) {ret[v - l] = v;}return ret;}
vi &range(int r) {return range(0, r);}
vi tov(vb& a){    vi ret;    rep(i,sz(a)){        if(a[i])ret.push_back(i);    }    return ret;}

#define rge range
#define upd update
//† ←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
#define VEC vector
vvc (ba);
ll N, M, H, W;
vi A, B, C;
/*@formatter:off*/
#undef getid
#undef getid_1
#undef getid_2

//#define type_id_graph
#ifdef type_id_graph
#define forg(gi, ve) for (ll gi = 0,forglim = ve.size(), f, t, c,ty,id; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c,ty=ve[gi].ty, id = ve[gi].id, true); ++gi)
#define fort(gi, ve) for (ll gi = 0,forglim = ve.size(); gi < forglim ; ++gi)
#define fort_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(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,forglim = ve.size(), f, t, c, ty, id; gi < forglim && (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_ {    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 :
    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) {}
    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) {}
    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
#define forg(gi, ve) for (ll gi = 0,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)
//#define fort(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(gi, ve) for (ll gi = 0,forglim = ve.size(); gi < forglim ; ++gi)
#define fort_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,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)

template<class T> struct edge_ {    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 :
    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) {}
    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) {}
    // 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> vi dis_path(digraph<T> &g, vector<T> &dis, int s, int t, T 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> vi dis_path(undigraph<T> &g, vector<T> &dis, int s, int t, T 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> vi dis_path(digraph<T> &g, vector<vector<T>> &dis, int s, int t, T init_value) { return dis_path(g, dis[s], s, t, init_value); }
template<class T> vi dis_path(undigraph<T> &g, vector<vector<T>> &dis, int s, int t, T init_value) { return dis_path(g, dis[s], s, t, init_value); }
//O(N^2)
vi to_vi(const int& i){return vi{i};}
vi to_vi(const vi& v){return v;}
/*@formatter:on*/
template<class T, class U> vector<T> dijkstra_mitu(graph<T> &g, const U &S_, T init_value) {
    vi S = to_vi(S_);
    int n = g.n;
    int initValue = MAX<int>();
    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> vector<T> dijkstra_01(graph<T> &g, const U &S_, T init_value) {
    int N = g.n;
    vi S = to_vi(S_);
    vector<T> dis(N, linf);
    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] == linf)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_; }
};
template<class Q, class T, class U> vector<T> private_dijkstra(graph<T> &g, U &S_, T 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(gi, 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> vector<T> dijkstra(graph<T> &g, U S_, T 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); }}
/*@formatter:off*/
//dijkstra_cou<mint> : 数える型で書く return vp(dis,cou)
template<class COU,class T=int> auto dijkstra_cou(const graph<T> &g, int s, T 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);}
//密グラフの時、warshallに投げる
template<class T> vector<vector<T>> dijkstra_all(const graph<T> &g, T 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;}
//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(gi, 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(gi, 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;}
template<class T> vector<vector<T>> warshall(graph<T> &g, T init_value) {    int n = g.n;    assert(n < 1e4);    vector<vector<T> > dis(n, vector<T>(n, linf));    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;}
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(gi, 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(gi, 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(i, 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(i, 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_; }
    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(gi, 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(gi, g[i]) {                if (t != par(i))q.push(t);            }        }        return res;    }int dfs_count_subtree(int p,int i){        childs[i] =1;        fort(gi, g[i]){fort_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);        }    }
    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(gi, 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(gi, g[u]) {                    if (t == par_[u]) {                        fedge(g[u][gi]);                        u = par_[u];                        break;                    }                }            }        vector<edge_<int>> qs;        while (v != r) {                forg(gi, 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 grid_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
    grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {}
    grid_k6(vector<vector<char>> ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])) {        rep(h, H) {            rep(w, W) {                if (ba[h][w] == wall)con;                int f = getid(h, w);                if (w + 1 < W && ba[h][w + 1] != wall) { add(f, getid(h, w + 1)); }                if (h + 1 < H && ba[h + 1][w] != wall) { add(f, getid(h + 1, w)); }            }        }    }
    /*o -> o, o -> x, x -> o, x-> x*/
    grid_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) {        rep(h, H) {            rep(w, W) {                add(h, w, h + 1, w);                add(h, w, h - 1, w);                add(h, w, h, w + 1);                add(h, w, h, w - 1);            }        }    }
    void add(int fh, int fw, int th, int tw) {
        if (ins(fh, fw, H, W) && ins(th, tw, H, W)) {
            int cm = 0;
            if (ba[fh][fw] == wall) { cm += 2; }
            if (ba[th][tw] == wall) { cm++; }
            int f = getid(fh, fw);
            int t = getid(th, tw);
            g[f].emplace_back(f, t, costs[cm]);
        }
    }
    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> {
public:
    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){}
    digrid_k6(vector<vector<char>> ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])) {        rep(h, H) {            rep(w, W) {                if (ba[h][w] == wall)con;                int f = getid(h, w);                if (w + 1 < W && ba[h][w + 1] != wall) { add(f, getid(h, w + 1)); }                if (h + 1 < H && ba[h + 1][w] != wall) { add(f, getid(h + 1, w)); }            }        }    }
    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); }
};
//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


//@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(gi, 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(gi, 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(gi, 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(gi, 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(gi, 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(gi, 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(gi, 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(gi, 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(gi, E[cur]) if (t != pre) {            P v = farthest____(E, t, cur, d + 1, 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, 1);    fora(v, per) { forg(gi, g[v]) { chma(dp[t], dp[f] + 1); }}    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(gi, 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(gi, 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(gi, g[v]) { 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(gi, 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(gi, 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> int krus(undigraph<T> &g) {    int res = 0;    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res += e.c; }}    return res;}
template<class T> vector<edge_<T>> krus_ed(undigraph<T> &g) {    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    vector<edge_<T>> res;    fora(e, E) { if (uf.unite(e.f, e.t)) { res.push_back(e); }}    return res;}
//森を返す
template<class T> tree<T> krus_un(undigraph<T> &g) {    undigraph<T> res(g.n);    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res.add(e.f, e.t); }}    return res;}
template<class T> tree<T> krus_tr(undigraph<T> &g) {    tree<T> res(g.n);    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res.add(e.f, e.t); }}    return res;}

template<class T> int krus( vector<edge_<T>> &g) {int n=0;    fora(e,g){chma(n,max(e.f,e.t)+1);};    int res = 0;    unionfind uf(n);    auto E = g;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res += e.c; }}    return res;}
template<class T> vector<edge_<T>> krus_ed( vector<edge_<T>> &g) {    int n=0;    fora(e,g){chma(n,max(e.f,e.t)+1);};    vector<edge_<T>> res;    unionfind uf(n);    auto E = g;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res += e; }}    return res;};
template<class T> vi krus_i( vector<edge_<T>> &g) {    int n=0;    fora(e,g){chma(n,max(e.f,e.t)+1);};    vi res;    unionfind uf(n);    auto E = g;    sort(E);    int i = 0;    fora(e, E) {        if (uf.unite(e.f, e.t)) { res += i; }        i++;    }    return res;}
//@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> vector<vector<int>> table(graph<T> &g,T init_value) {    vvi(res, g.n, g.n,init_value);    rep(i, g.n) { forg(gi, 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(gi, 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(gi, g[i]) { chma(tn, ty); }}    vector<vector<edge_<T>>> res(tn + 1);    vi was(g.n);    rep(i, g.n) {        forg(gi, 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
/*@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() {
    D s, t;
    in(s, t);
    VEC<R> A;
    in(N);
    na(A, N);
    A = R(s, 0) + A;
    A += R(t, 0);
    N += 2;
    undigraph<double> g(2 * k5);
    rep(i, N) {
        rep(j, N) {
            auto v = dis(A[i].o, A[j].o);
            auto d = max(0, v - A[i].r - A[j].r);
            g.add(i, j, d);
        }
    }
    auto D = dijkstra(g, 0, (epsdou)linf);
    deb(D);
    out(D[N - 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) {
//        cout << "完璧 : solveを書き直そう" << endl;
//        cout << "     : そして、solve()を呼び出すのだ" << endl;
//        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
    }
    if (was_deb && sz(res_mes)) {
        cerr << "result = " << endl << res_mes << endl;
    }
    if (sz(message)) {
        cerr << "****************************" << endl;
        cerr << "Note." << endl;
        cerr << message << endl;
        cerr << "****************************" << endl;
    }
#endif
    return 0;
};


Measured memory_bytes: 153436