// Slower C++ solution:
//todo 文字数を少なくする

//mintより高速に計算したい時用に modmul() modadd (正の値限定) を作る

//#pragma GCC optimize ("-O3")

#include <bits/stdc++.h>

using namespace std;

//@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));

    };

} __initon;

//衝突対策

#define ws ___ws



//@必須構造

struct T {

    int f, s, t;

    T() { f = -1, s = -1, t = -1; }

    T(int f, int s, int t) : f(f), s(s), t(t) {}

    bool operator<(const T &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; 大きい順

    }

    bool operator>(const T &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 T &r) const {

        return f == r.f && s == r.s && t == r.t;

    }

    bool operator!=(const T &r) const {

        return f != r.f || s != r.s || t != r.t;

    }

    int operator[](int i) {

        assert(i < 3);

        return i == 0 ? f : i == 1 ? s : t;

    }

};

struct F {

    int a, b, c, d;

    F() { a = -1, b = -1, c = -1, d = -1; }

    F(int a, int b, int c, int d) : a(a), b(b), c(c), d(d) {}

    bool operator<(const F &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 F &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 F &r) const {

        return a == r.a && b == r.b && c == r.c && d == r.d;

    }

    bool operator!=(const F &r) const {

        return a != r.a || b != r.b || c != r.c || d != r.d;

    }

    int operator[](int i) {

        assert(i < 4);

        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;

    }

};



T mt(int a, int b, int c) {

    return T(a, b, c);

}

//@マクロ省略系 型,構造



#define int long long

#define ll long long

#define double long double

#define ull unsigned long long

using dou = double;

using itn = int;

using str = string;

using bo= bool;

#define au auto

using P = pair<ll, ll>;

using pd =pair<dou, dou>;

#define fi first

#define se second

#define vec vector

#define beg begin

#define rbeg rbegin

#define con continue

#define bre break

#define brk break

#define is ==



#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))



//マクロ省略系 コンテナ



using vi = vector<int>;

#define _overloadvvi(_1, _2, _3, _4, name, ...) name

#define vvi0() vec<vi>

#define vvi1(a) vec<vi> a

#define vvi2(a, b) vec<vi> a(b)

#define vvi3(a, b, c) vec<vi> a(b,vi(c))

#define vvi4(a, b, c, d) vec<vi> a(b,vi(c,d))

#define vvi(...) _overloadvvi(__VA_ARGS__,vvi4,vvi3,vvi2 ,vvi1,vvi0)(__VA_ARGS__)





using vl = vector<ll>;

#define _overloadvvl(_1, _2, _3, _4, name, ...) name

#define vvl1(a) vec<vl> a

#define vvl2(a, b) vec<vl> a(b)

#define vvl3(a, b, c) vec<vl> a(b,vl(c))

#define vvl4(a, b, c, d) vec<vl> a(b,vl(c,d))

#define vvl(...) _overloadvvl(__VA_ARGS__,vvl4,vvl3,vvl2 ,vvl1)(__VA_ARGS__)



using vb = vector<bool>;

#define _overloadvvb(_1, _2, _3, _4, name, ...) name

#define vvb1(a) vec<vb> a

#define vvb2(a, b) vec<vb> a(b)

#define vvb3(a, b, c) vec<vb> a(b,vb(c))

#define vvb4(a, b, c, d) vec<vb> a(b,vb(c,d))

#define vvb(...) _overloadvvb(__VA_ARGS__,vvb4,vvb3,vvb2 ,vvb1)(__VA_ARGS__)



using vs = vector<string>;

#define _overloadvvs(_1, _2, _3, _4, name, ...) name

#define vvs1(a) vec<vs> a

#define vvs2(a, b) vec<vs> a(b)

#define vvs3(a, b, c) vec<vs> a(b,vs(c))

#define vvs4(a, b, c, d) vec<vs> a(b,vs(c,d))

#define vvs(...) _overloadvvs(__VA_ARGS__,vvs4,vvs3,vvs2 ,vvs1)(__VA_ARGS__)



using vd = vector<double>;

#define _overloadvvd(_1, _2, _3, _4, name, ...) name

#define vvd1(a) vec<vd> a

#define vvd2(a, b) vec<vd> a(b)

#define vvd3(a, b, c) vec<vd> a(b,vd(c))

#define vvd4(a, b, c, d) vec<vd> a(b,vd(c,d))

#define vvd(...) _overloadvvd(__VA_ARGS__,vvd4,vvd3,vvd2 ,vvd1)(__VA_ARGS__)



using vc=vector<char>;

#define _overloadvvc(_1, _2, _3, _4, name, ...) name

#define vvc1(a) vec<vc> a

#define vvc2(a, b) vec<vc> a(b)

#define vvc3(a, b, c) vec<vc> a(b,vc(c))

#define vvc4(a, b, c, d) vec<vc> a(b,vc(c,d))

#define vvc(...) _overloadvvc(__VA_ARGS__,vvc4,vvc3,vvc2 ,vvc1)(__VA_ARGS__)



using vp = vector<P>;

#define _overloadvvp(_1, _2, _3, _4, name, ...) name

#define vvp1(a) vec<vp> a

#define vvp2(a, b) vec<vp> a(b)

#define vvp3(a, b, c) vec<vp> a(b,vp(c))

#define vvp4(a, b, c, d) vec<vp> a(b,vp(c,d))

#define vvp(...) _overloadvvp(__VA_ARGS__,vvp4,vvp3,vvp2 ,vvp1)(__VA_ARGS__)



using vt = vector<T>;

#define _overloadvvt(_1, _2, _3, _4, name, ...) name

#define vvt1(a) vec<vt> a

#define vvt2(a, b) vec<vt> a(b)

#define vvt3(a, b, c) vec<vt> a(b,vt(c))

#define vvt4(a, b, c, d) vec<vt> a(b,vt(c,d))

#define vvt(...) _overloadvvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1)(__VA_ARGS__)



#define v3i(a, b, c, d) vector<vector<vi>> a(b, vector<vi>(c, vi(d)))

#define v3d(a, b, c, d) vector<vector<vd>> a(b, vector<vd>(c, vd(d)))

#define v3m(a, b, c, d) vector<vector<vm>> a(b, vector<vm>(c, vm(d)))



#define _vvi vector<vi>

#define _vvl vector<vl>

#define _vvb vector<vb>

#define _vvs vector<vs>

#define _vvd vector<vd>

#define _vvc vector<vc>

#define _vvp vector<vp>



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<int>(__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 PQ priority_queue<ll, vector<ll>, greater<ll> >

#define tos to_string

using mapi = map<int, int>;

using mapp = map<P, int>;

using mapd = map<dou, int>;

using mapc = map<char, int>;

using maps = map<str, int>;

using seti = set<int>;

using setd = set<dou>;

using setc = set<char>;

using sets = set<str>;

using qui = queue<int>;

#define bset bitset

#define uset unordered_set

#define mset multiset

#define umap unordered_map

#define umapi unordered_map<int,int>

#define umapp unordered_map<P,int>

#define mmap multimap



//マクロ 繰り返し

#define _overloadrep(_1, _2, _3, _4, name, ...) name

# define rep1(n) for(int rep1i = 0,_lim=n; rep1i < _lim ; rep1i++)

# define _rep(i, n) for(int i = 0,_lim=n; i < _lim ; i++)

#define repi(i, m, n) for(int i = m,_lim=n; i < _lim ; i++)

#define repadd(i, m, n, ad) for(int i = m,_lim=n; i < _lim ; i+= ad)

#define rep(...) _overloadrep(__VA_ARGS__,repadd,repi,_rep,rep1)(__VA_ARGS__)

#define _rer(i, n) for(int i = n; i >= 0 ; i--)

#define reri(i, m, n) for(int i = m,_lim=n; i >= _lim ; i--)

#define rerdec(i, m, n, dec) for(int i = m,_lim=n; i >= _lim ; i-=dec)

#define rer(...) _overloadrep(__VA_ARGS__,rerdec,reri,_rer,)(__VA_ARGS__)



#define reps2(i, j, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)

#define reps3(i, j, k, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)

#define reps4(i, j, k, l, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)for(int l = 0; l < _lim ; l++)



#define _overloadreps(_1, _2, _3, _4, _5, name, ...) name

#define reps(...) _overloadreps(__VA_ARGS__,reps4,reps3,reps2,_rep,)(__VA_ARGS__)



#define repss(i, j, k, a, b, c) for(int i = 0; i < a ; i++)for(int j = 0; j < b ; j++)for(int k = 0; k < c ; k++)



#define fora(a, b) for(auto&& a : b)

#define forg(gi, ve) for (int gi = 0,_lim = ve.size(), f, t, c; gi < _lim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)

#define fort(gi, ve) for (int gi = 0, f, t, c; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)if(t!=p)

//#define fort(gi, ve) for (int gi = 0, f, t, c;gi<ve.size()&& (gi+= (ve[gi].t==p))< ve.size() && (f = ve[gi].f,t=ve[gi].t, c = ve[gi].c,true); gi++)



#define form(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it).fi < r; it++)

#define forit(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r; it++)



//マクロ 定数

#define k3 1010

#define k4 10101

#define k5 101010

#define k6 1010101

#define k7 10101010

const int inf = (int) 1e9 + 100;

const ll linf = (ll) 1e18 + 100;

const char infc = '{';

const string infs = "{";

const double eps = 1e-9;

const double PI = 3.1415926535897932384626433832795029L;

ll ma = numeric_limits<ll>::min();

ll mi = numeric_limits<ll>::max();

const int y4[] = {-1, 1, 0, 0};

const int x4[] = {0, 0, -1, 1};

const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};

const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};



//マクロ省略形 関数等

#define arsz(a) (sizeof(a)/sizeof(a[0]))

#define sz(a) ((int)(a).size())

//#define rs resize

#define mp make_pair

#define pb push_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(int a) { return !(a & 1); }

constexpr bool od(int 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);

        }

    };

}

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 << endl;

    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 (int 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 (int i = 0; i < vec.size(); i++) {

        for (int j = 0; j < vec[0].size(); j++) {

            os << vec[i][j];

        }

        os << endl;

    }

    return os;

}

template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {

    for (auto &&v:m) os << v;

    return os;

}



template<typename V, typename H> void resize(vector<V> &vec,

        const H head

) {

    vec.

            resize(head);

}

template<typename V, typename H, typename ... T> void resize(vector<V> &vec,

        const H &head,

        const T ... tail

) {

    vec.

            resize(head);

    for (

        auto &v

            : vec)

        resize(v, tail

                ...);

}

template<typename T, typename _Pr> bool all_of(const vector<T> &vec, _Pr pred) { return std::all_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> bool any_of(const vector<T> &vec, _Pr pred) { return std::any_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> bool none_of(const vector<T> &vec, _Pr pred) { return std::none_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> const typename vector<T>::const_iterator find_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred); }

template<class T> bool contains(const std::string &s, const T &v) { return s.find(v) != std::string::npos; }

template<typename T> bool contains(const vector<T> &vec, const T &val) { return std::find(vec.begin(), vec.end(), val) != vec.end(); }

template<typename T, typename _Pr> bool contains_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred) != vec.end(); }

template<class T, class U> void replace(vector<T> &a, T

key,

        U v

) {

    replace(a

                    .

                            begin(), a

                    .

                            end(), key, v

    );

}

void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }

void replace(str &a, str key, str v) {

    if (v == "")a.erase(remove(all(a), key[0]), a.end());

    else replace(all(a), key[0], v[0]);

}

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, '#'); }



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)); }



template<class T> T pop(set<T> &set) {

    T res = *set.begin();

    set.erase(set.find(res));

    return res;

}

template<class T> T pop(mset<T> &set) {

    T res = *set.begin();

    set.erase(set.find(res));

    return res;

}

template<class T> T popBack(set<T> &set) {

    T res = *set.rbegin();

    set.erase(set.find(res));

    return res;

}

template<class T> T popBack(mset<T> &set) {

    T res = *set.rbegin();

    set.erase(set.find(res));

    return res;

}

inline void sort(string &a) { sort(a.begin(), a.end()); }

template<class T> inline void sort(vector<T> &a) { sort(a.begin(), a.end()); }

template<class T> inline void sort(vector<T> &a, int

len) {

    sort(a

                 .

                         begin(), a

                                          .

                                                  begin()

                                  + len);

}

template<class T, class F> inline void sort(vector<T> &a, F

f) {

    sort(a

                 .

                         begin(), a

                 .

                         end(),

         [&](

                 T a, T

         b) {

             return

                     f(a)

                     <

                     f(b);

         });

}

enum ___pcomparator {

    fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd

};

inline void sort(vector<P> &a, ___pcomparator

type) {

    switch (type) {

        case fisi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi < r.fi : l.se < r.

                             se;

                 });

            break;

        case fisd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi < r.fi : l.se > r.

                             se;

                 });

            break;

        case fdsi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi > r.fi : l.se < r.

                             se;

                 });

            break;

        case fdsd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi > r.fi : l.se > r.

                             se;

                 });

            break;

        case sifi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se < r.se : l.fi < r.

                             fi;

                 });

            break;

        case sifd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se < r.se : l.fi > r.

                             fi;

                 });

            break;

        case sdfi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se > r.se : l.fi < r.

                             fi;

                 });

            break;

        case sdfd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se > r.se : l.fi > r.

                             fi;

                 });

            break;

    }

};

inline void sort(vector<T> &a, ___pcomparator

type) {

    switch (type) {

        case fisi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f < r.f : l.s < r.

                             s;

                 });

            break;

        case

            fisd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f < r.f : l.s > r.

                             s;

                 });

            break;

        case

            fdsi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f > r.f : l.s < r.

                             s;

                 });

            break;

        case

            fdsd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f > r.f : l.s > r.

                             s;

                 });

            break;

        case

            sifi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s < r.s : l.f < r.

                             f;

                 });

            break;

        case

            sifd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s < r.s : l.f > r.

                             f;

                 });

            break;

        case

            sdfi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s > r.s : l.f < r.

                             f;

                 });

            break;

        case

            sdfd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s > r.s : l.f > r.

                             f;

                 });

            break;

    }

};

template<class T> inline void rsort(vector<T> &a) { sort(a.begin(), a.end(), greater<T>()); };

template<class T> inline void rsort(vector<T> &a, int

len) {

    sort(a

                 .

                         begin(), a

                                          .

                                                  begin()

                                  + len,

         greater<T>()

    );

};

template<class U, class F> inline void rsort(vector<U> &a, F

f) {

    sort(a

                 .

                         begin(), a

                 .

                         end(),

         [&](

                 U l, U

         r) {

             return

                     f(l)

                     >

                     f(r);

         });

};

template<class U> inline void sortp(vector<U> &a, vector<U> &b) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.eb(a[i], b[i]);

    sort(c);

    rep(i, n) {

        a[i] = c[i].first;

        b[i] = c[i].second;;

    }

};

//F = T<T>

//例えばreturn p.fi + p.se;

template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, F

f) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    sort(c, f

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, char

type) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    sort(c, type

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U> inline void rsortp(vector<U> &a, vector<U> &b) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.eb(a[i], b[i]);

    rsort(c);

    rep(i, n) {

        a[i] = c[i].first;

        b[i] = c[i].second;

    }

};

template<class U, class F> inline void rsortp(vector<U> &a, vector<U> &b, F

f) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    rsort(c, f

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.eb(a[i], b[i], c[i]);

    sort(r);

    rep(i, n) {

        a[i] = r[i].f;

        b[i] = r[i].s;

        c[i] = r[i].t;

    }

};

template<class U, class F> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c, F

f) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.

                eb(a[i], b[i], c[i]

        );

    sort(r, f

    );

    rep(i, n) {

        a[i] = r[i].

                f;

        b[i] = r[i].

                s;

        c[i] = r[i].

                t;

    }

};

template<class U, class F> inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c, F

f) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.

                eb(a[i], b[i], c[i]

        );

    rsort(r, f

    );

    rep(i, n) {

        a[i] = r[i].

                f;

        b[i] = r[i].

                s;

        c[i] = r[i].

                t;

    }

};

template<class U> inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.eb(a[i], b[i], c[i]);

    rsort(r);

    rep(i, n) {

        a[i] = r[i].f;

        b[i] = r[i].s;

        c[i] = r[i].t;

    }

};



//indexをつけてsortする (second)

template<class T> auto sorti(vector<T> &a) {

    int n = sz(a);

    vp ret(n);

    rep(i, n)ret[i] = P(a[i], i);

    sort(ret);

    return ret;

};

template<class T> auto rsorti(vector<T> &a) {

    int n = sz(a);

    vp ret(n);

    rep(i, n)ret[i] = P(a[i], i);

    rsort(ret);

    return ret;

};

template<class T> inline void sort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; i++)sort(a[i]); }

template<class T> inline void rsort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; i++)rsort(a[i]); }

template<typename A, size_t N, typename T> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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 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;

}



template<typename V, typename T>

void fill(V &xx, const T vall) {

    xx = vall;

}

template<typename V, typename T>

void fill(vector<V> &vecc,

        const T vall

) {

    for (

        auto &&vx

            : vecc)

        fill(vx, vall

        );

}



//@汎用便利関数 入力

inline int in() {

    int ret;

//    scanf("%lld", &ret);

    cin >> ret;

    return ret;

}

inline string sin() {

    string ret;

    cin >> ret;

    return ret;

}

template<class T> inline void in(T &head) { cin >> head; }

template<class T, class... U> inline void in(T &head, U &... tail) {

    cin >> head;

    in(tail...);

}



#define _overloaddin(_1, _2, _3, _4, name, ...) name

#define din1(a) int a;cin>>a

#define din2(a, b) int a,b;cin>>a>> b

#define din3(a, b, c) int a,b,c;cin>>a>>b>>c

#define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d

#define din(...) _overloaddin(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)



#define _overloaddind(_1, _2, _3, _4, name, ...) name

#define din1d(a) din1(a);a--

#define din2d(a, b) din2(a,b);a--,b--

#define din3d(a, b, c) din3(a,b,c);a--,b--,c--

#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--

#define dind(...) _overloaddind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)





#define _overloadout(_1, _2, _3, _4, name, ...) name

#define out1(a) cout<<a<<endl

#define out2(a, b) cout<<a<<" "<< b<<endl

#define out3(a, b, c) cout<<a<<" "<<b<<" "<<c<<endl

#define out4(a, b, c, d) cout<<a<<" "<<b<<" "<<c<<" "<<d<<endl

#define out(...) _overloadout((__VA_ARGS__),out4,out3,out2,out1)((__VA_ARGS__))



template<class T> void outl(vector<T> &a) { fora(v, a)cout << v << endl; }

#define na(a, n) a.resize(n); rep(_i,n) cin >> a[_i];

#define dna(a, n) vi a(n); rep(_i,n) cin >> a[_i];

#define nao(a, n) a.resize(n+1); rep(_i,n) cin >> a[_i+1];

#define nad(a, n) a.resize(n); rep(_i,n){ cin >> a[_i]; a[_i]--;}

#define na2(a, b, n) a.resize(n),b.resize(n);rep(_i, n)cin >> a[_i] >> b[_i];

#define dna2(a, b, n) vi a(n),b(n);rep(_i, n)cin >> a[_i] >> b[_i];

#define nao2(a, b, n) a.resize(n+1),b.resize(n+1);rep(_i, n)cin >> a[_i+1] >> b[_i+1];

#define na2d(a, b, n) a.resize(n),b.resize(n);rep(_i, n){cin >> a[_i] >> b[_i];a[_i]--,b[_i]--;}

#define na3(a, b, c, 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(_i, n)cin >> a[_i] >> b[_i] >> c[_i];

#define na3d(a, b, c, 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(_i, n){cin >> a[_i] >> b[_i] >> c[_i];a[_i]--,b[_i]--,c[_i]--;}

#define nt(a, h, w) resize(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi];

#define ntd(a, h, w) rs(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi], a[_hi][_wi]--;

#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(_hi,1,h+1)rep(_wi,1,w+1) cin >> a[_hi][_wi];



//デバッグ

#define sp << " " <<



#define debugName(VariableName) # VariableName



#define _deb1(x)  debugName(x)<<" = "<<x

#define _deb2(x, ...) _deb1(x) <<", "<< _deb1(__VA_ARGS__)

#define _deb3(x, ...) _deb1(x) <<", "<< _deb2(__VA_ARGS__)

#define _deb4(x, ...) _deb1(x) <<", "<< _deb3(__VA_ARGS__)

#define _deb5(x, ...) _deb1(x) <<", "<< _deb4(__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__)



#define _overloadebug(_1, _2, _3, _4, _5, _6, _7, _8, _9, _10, name, ...) name



#ifdef _DEBUG

#define deb(...)  cerr<< _overloadebug(__VA_ARGS__,_deb10,_deb9,_deb8,_deb7,_deb6,_deb5,_deb4,_deb3,_deb2,_deb1)(__VA_ARGS__) <<endl

#else

#define deb(...) ;

#endif



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



//よく使うクラス、構造体

class UnionFind {

public:

    vi par, rank, sizes;

    int n, trees;

    UnionFind(int n) : n(n), trees(n) {

        par.resize(n), rank.resize(n), sizes.resize(n);

        rep(i, n)par[i] = i, sizes[i] = 1;

    }

    int root(int x) {

        if (par[x] == x)return x;

        else return par[x] = root(par[x]);

    }

    int find(int x) { return root(x); }

    void unite(int x, int y) {

        x = root(x);

        y = root(y);

        if (x == y)return;

        if (rank[x] < rank[y])swap(x, y);

        trees--;

        par[y] = x;

        sizes[x] += sizes[y];

        if (rank[x] == rank[y])rank[x]++;

    }

    bool same(int x, int y) { return root(x) == root(y); }

    int size(int x) { return sizes[root(x)]; }

    //順不同 umapなので

    vec<vi> sets() {

        vec<vi> res(trees);

        umap<int, vi> map;

        rep(i, n) map[root(i)].push_back(i);

        int i = 0;

        for (auto &&p:map) {

            int r = p.fi;

            res[i].push_back(r);

            for (auto &&v:p.se) {

                if (r == v)continue;

                res[i].push_back(v);

            }

            i++;

        }

        return res;

    }

};



using bint =__int128;

using u32 = unsigned int;

using u64 = unsigned long long;

using u128 = __uint128_t;

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 = '-';

        }

        int len = std::end(buffer) - d;

        if (dest.rdbuf()->sputn(d, len) != len) {

            dest.setstate(std::ios_base::badbit);

        }

    }

    return dest;

}



__int128 toi128(string &s) {

    __int128 ret = 0;

    for (int i = 0; i < s.length(); i++)

        if ('0' <= s[i] && s[i] <= '9')

            ret = 10 * ret + s[i] - '0';

    return ret;

}





template<typename T> T minv(T a, T m);

template<typename T> T minv(T a);



template<typename T>

class Modular {

public:

    using Type = typename decay<decltype(T::value)>::type;



    constexpr Modular() : value() {}

    template<typename U>

    Modular(const U &x) {

        value = normalize(x);

    }



    template<typename U>

    static Type normalize(const U &x) {

        Type v;

        if (-mod() <= x && x < mod()) v = static_cast<Type>(x);

        else v = static_cast<Type>(x % mod());

        if (v < 0) v += mod();

        return v;

    }



    const Type &operator()() const { return value; }

    template<typename U>

    explicit operator U() const { return static_cast<U>(value); }

    constexpr static Type mod() { return T::value; }



    Modular &operator+=(const Modular &other) {

        if ((value += other.value) >= mod()) value -= mod();

        return *this;

    }

    Modular &operator-=(const Modular &other) {

        if ((value -= other.value) < 0) value += mod();

        return *this;

    }

    template<typename U> Modular &operator+=(const U &other) { return *this += Modular(other); }

    template<typename U> Modular &operator-=(const U &other) { return *this -= Modular(other); }

    Modular &operator++() { return *this += 1; }

    Modular &operator--() { return *this -= 1; }

    Modular operator++(signed) {

        Modular result(*this);

        *this += 1;

        return result;

    }

    Modular operator--(signed) {

        Modular result(*this);

        *this -= 1;

        return result;

    }

    Modular operator-() const { return Modular(-value); }



    template<typename U = T>

    typename enable_if<is_same<typename Modular<U>::Type, signed>::value, Modular>::type &operator*=(const Modular &rhs) {

#ifdef _WIN32

        uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);

        uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;

        asm(

        "divl %4; \n\t"

        : "=a" (d), "=d" (m)

        : "d" (xh), "a" (xl), "r" (mod())

        );

        value = m;

#else

        value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));

#endif

        return *this;

    }

    template<typename U = T>

    typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type &operator*=(const Modular &rhs) {

        int64_t q = static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());

        value = normalize(value * rhs.value - q * mod());

        return *this;

    }

    template<typename U = T>

    typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type &operator*=(const Modular &rhs) {

        value = normalize(value * rhs.value);

        return *this;

    }

    Modular &operator/=(const Modular &other) { return *this *= Modular(minv(other.value)); }

    template<typename U> friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);

    template<typename U> friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);

    template<typename U> friend std::istream &operator>>(std::istream &stream, Modular<U> &number);

    operator int() { return normalize(value); }

private:

    Type value;

};



template<typename T> bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value == rhs.value; }

template<typename T, typename U> bool operator==(const Modular<T> &lhs, U rhs) { return lhs == Modular<T>(rhs); }

template<typename T, typename U> bool operator==(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) == rhs; }



template<typename T> bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) { return !(lhs == rhs); }

template<typename T, typename U> bool operator!=(const Modular<T> &lhs, U rhs) { return !(lhs == rhs); }

template<typename T, typename U> bool operator!=(U lhs, const Modular<T> &rhs) { return !(lhs == rhs); }



template<typename T> bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value < rhs.value; }



template<typename T> Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }

template<typename T, typename U> Modular<T> operator+(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) += rhs; }

template<typename T, typename U> Modular<T> operator+(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }



template<typename T> Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }

template<typename T, typename U> Modular<T> operator-(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) -= rhs; }

template<typename T, typename U> Modular<T> operator-(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }



template<typename T> Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }

template<typename T, typename U> Modular<T> operator*(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) *= rhs; }

template<typename T, typename U> Modular<T> operator*(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }



template<typename T> Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U> Modular<T> operator/(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U> Modular<T> operator/(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }



constexpr signed MOD = 1e9 + 7;//MOD

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



constexpr int mint_len = 1400001;

vi fac, finv, inv;

vi p2;

void setmod() {

//    p2.resize(mint_len);p[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i] = p2[i - 1] * 2 % MOD;

    fac.resize(mint_len);

    finv.resize(mint_len);

    inv.resize(mint_len);

    inv[1] = fac[0] = finv[0] = 1;

    for (int i = 2; i < mint_len; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;

    for (int i = 1; i < mint_len; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;

}

#define smod setmod

inline mint com(int n, int r) {

    if (r < 0 || r > n) return 0;

    return mint(finv[r] * fac[n] % MOD * finv[n - r]);

}

inline mint pom(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return mint(fac[n] * finv[n - 1]);

}

inline mint npr(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return mint(fac[n] * finv[n - r]);

}

inline int nprin(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return fac[n] * finv[n - r] % MOD;

}



inline int icom(int n, int r) {

    const int NUM_ = 1400001;

    static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];

    if (fac[0] == 0) {

        inv[1] = fac[0] = finv[0] = 1;

        for (int i = 2; i <= NUM_; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;

        for (int i = 1; i <= NUM_; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;

    }

    if (r < 0 || r > n) return 0;

    return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;

}

#define ncr com

#define ncri icom

inline mint nhr(int n, int r) { return com(n + r - 1, r); }

inline mint hom(int n, int r) { return com(n + r - 1, r); }

inline int nhri(int n, int r) { return icom(n + r - 1, r); }





template<typename T> inline T minv(T a, T m) {

    T u = 0, v = 1;

    while (a != 0) {

        T t = m / a;

        m -= t * a;

        swap(a, m);

        u -= t * v;

        swap(u, v);

    }

    assert(m == 1);

    return u;

}

template<typename T> inline T minv(T a) {

    if (a < mint_len)return inv[a];

    T u = 0, v = 1;

    T m = MOD;

    while (a != 0) {

        T t = m / a;

        m -= t * a;

        swap(a, m);

        u -= t * v;

        swap(u, v);

    }

    assert(m == 1);

    return u;

}



template<typename T, typename U> inline Modular<T> mpow(const Modular<T> &a, const U &b) {

    assert(b >= 0);

    int x = a(), res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= MOD;

        (x *= x) %= MOD;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U, typename V> inline mint mpow(const T a, const U b, const V m = MOD) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= m;

        (x *= x) %= m;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U> inline mint mpow(const T a, const U b) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= MOD;

        (x *= x) %= MOD;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U, typename V> inline int mpowi(const T &a, const U &b, const V &m = MOD) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= m;

        (x *= x) %= m;

        p >>= 1;

    }

    return res;

}

template<typename T> inline string to_string(const Modular<T> &number) {

    return to_string(number());

}



template<typename T> std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {

    return stream << number();

}



template<typename T> std::istream &operator>>(std::istream &stream, Modular<T> &number) {

    typename common_type<typename Modular<T>::Type, int64_t>::type x;

    stream >> x;

    number.value = Modular<T>::normalize(x);

    return stream;

}



using PM = pair<mint, mint>;

using vm = vector<mint>;

using mapm = map<int, mint>;

using umapm = umap<int, mint>;

#define _overloadvvm(_1, _2, _3, _4, name, ...) name

#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)



#define vvm1(a) vec<vm> a

#define vvm2(a, b) vec<vm> a(b)

#define vvm3(a, b, c) vec<vm> a(b,vm(c))

#define vvm4(a, b, c, d) vec<vm> a(b,vm(c,d))

#define vvm(...) _overloadvvm(__VA_ARGS__,vvm4,vvm3,vvm2 ,vvm1)(__VA_ARGS__)





//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); }

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



//便利関数



//テスト用

char ranc() {

    return (char) ('a' + rand() % 26);

}

int rand(int min, int 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);

        }

    }

}

vi ranv(int n, int min, int max) {

    vi v(n);

    rep(i, n)v[i] = rand(min, max);

    return v;

}

//単調増加

vi ranvi(int n, int min, int 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;

}

void ranvlr(int n, int min, int 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]);

    }

}

//便利 汎用

vp run_length(vi &a) {

    vp 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, int>> run_length(string &a) {

    vector<pair<char, 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;

}

template<class F> inline int mgr(int ok, int ng, F f) {

#define _mgrbody int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid;

    if (ok < ng)while (ng - ok > 1) { _mgrbody } else while (ok - ng > 1) { _mgrbody }

    return ok;

}



template<class F> inline int mgr(int ok, int ng, int second, F f) {

#define _mgrbody2 int mid = (ok + ng) / 2; if (f(mid, second))ok = mid; else ng = mid;

    if (ok < ng) while (ng - ok > 1) { _mgrbody2 } else while (ok - ng > 1) { _mgrbody2 }

    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 {

        rep(i, sz(a)) {

            if (a[i] < b[i]) {

                return b;

            } else if (a[i] > b[i])return a;

        }

    }

    return a;

}

//strを整数として比較

string smin(str &a, str b) {

    if (sz(a) < sz(b)) {

        return a;

    } else if (sz(a) > sz(b)) {

        return b;

    } else {

        rep(i, sz(a)) {

            if (a[i] < b[i]) {

                return a;

            } else if (a[i] > b[i])return b;

        }

    }

    return a;

}

template<typename V, typename T> int find(vector<V> &a,

        const T key

) {

    rep(i, sz(a))if (a[i] == key)

            return

                    i;

    return -1;

}

template<typename V, typename T> P find(vector<vector<V >> &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);

}

template<typename V, typename U> T find(vector<vector<vector<V >>> &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);

}



template<typename V, typename T> int count2(V &a, const T k) { return a == k; }

template<typename V, typename T> int count2(vector<V> &a, const T k) {

    int ret = 0;

    fora(v, a)ret +=

                      count(v, k

                      );

    return

            ret;

}

template<typename V, typename T> int count(vector<V> &a, const T k) {

    int ret = 0;

    fora(v, a)ret +=

                      count2(v, k

                      );

    return

            ret;

}

int count(str a, str k) {

    int ret = 0, len = k.length();

    auto pos = a.find(k);

    while (pos != string::npos)pos = a.find(k, pos + len), ret++;

    return ret;

}

template<typename V> int count_odd(V &a) { return a % 2; }

template<typename V> int count_odd(vector<V> &a) {

    int ret = 0;

    fora(v, a)ret += count_odd(v);

    return ret;

}

template<typename V> int count_even(V &a) { return a % 2 == 0; }

template<typename V> int count_even(vector<V> &a) {

    int ret = 0;

    fora(v, a)ret += count_even(v);

    return ret;

}

//algorythm

void iota(vector<int> &ve, int

s,

        int n

) {

    ve.

            resize(n);

    iota(all(ve), s

    );

}

vi iota(int s, int len) {

    vi ve(len);

    iota(all(ve), s);

    return ve;

}



inline ll rev(ll a) {

    ll res = 0;

    while (a) {

        res *= 10;

        res += a % 10;

        a /= 10;

    }

    return res;

}

template<class T> void rev(vector<T> &a) {

    reverse(all(a));

}

template<class T> vector<T> revv(vector<T> &a) {

    vector<T> b = a;

    reverse(all(b));

    return b;

}

void inline rev(string &a) {

    reverse(all(a));

}

constexpr int p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};



int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }



int keta(ll v) { if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) 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 if (v < p10[15])return 15; else return 16; } else { if (v < p10[17])return 17; else return 18; }}}}



int dsum(int v) {

    int ret = 0;

    for (; v; v /= 10)ret += v % 10;

    return ret;

}

inline int sumd(int v) { return dsum(v); }



struct sint {

    int v;

    sint(int v) : v(v) {}

    operator int() { return v; }

    int operator[](int i) { return (v / p10[i]) % 10; }

    int top(int i) {

        int len = keta(v);

        return operator[](len - 1 - i);

    }

    //先頭からi番目にセット

    int top(int i, int k) {

        int len = keta(v);

        return set(len - 1 - i, k);

    }

    int set(int i, int k) {

        if (i < 0)return top(abs(i) - 1, k);

        return v += p10[i] * (k - (v / p10[i]) % 10);

    }

    int inc(int i, int k = 1) {

        return v += p10[i] * k;

    }

    int inct(int i, int k = 1) {

        return v += p10[keta(v) - i - 1] * k;

    }

    int dec(int i, int k = 1) {

        return v -= p10[i] * k;

    }

    int dect(int i, int k = 1) {

        return v -= p10[keta(v) - i - 1] * k;

    }

#define op(t, o)template<class T>inline t operator o(T r){return v o r;}

    op(int, +=);

    op(int, -=);

    op(int, *=);

    op(int, /=);

    op(int, %=);

    op(int, +);

    op(int, -);

    op(int, *);

    op(int, /);

    op(int, %);

    op(bool, ==);

    op(bool, !=);

    op(bool, <);

    op(bool, <=);

    op(bool, >);

    op(bool, >=);

#undef op

    template<class T> inline int operator<<=(T r) { return v *= p10[r]; }

    template<class T> inline int operator<<(T r) { return v * p10[r]; }

    template<class T> inline int operator>>=(T r) { return v /= p10[r]; }

    template<class T> inline int operator>>(T r) { return v / p10[r]; }

};

int mask10(int v) { return p10[v] - 1; }

//変換系

template<class T, class U> inline vector<T> keys(vector<pair<T, U>> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}

template<class T, class U> inline vector<U> keys(map<T, U> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}



template<class T, class U> inline vector<U> keys(umap<T, U> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}



template<class T, class U> vector<U> inline values(vector<pair<T, U>> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline vector<T> values(map<T, U> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline vector<T> values(umap<T, U> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline bool chma(T &a, const U &b) {

    if (a < b) {

        a = b;

        return true;

    }

    return false;

}

template<class U> inline bool chma(const U &b) { return chma(ma, b); }

template<class T, class U> inline bool chmi(T &a, const U &b) {

    if (b < a) {

        a = b;

        return true;

    }

    return false;

}

template<class U> inline bool chmi(const U &b) { return chmi(mi, b); }

template<class T> inline T min(T a, signed b) { return a < b ? a : b; }

template<class T> inline T max(T a, signed b) { return a < b ? b : a; }

template<class T> inline T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }

template<class T> inline T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }

template<class T> inline T min(vector<T> a) { return *min_element(all(a)); }

template<class T> inline T mini(vector<T> a) { return min_element(all(a)) - a.begin(); }

template<class T> inline T min(vector<T> a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T min(vector<T> a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a) { return *max_element(all(a)); }

template<class T> inline T maxi(vector<T> a) { return max_element(all(a)) - a.begin(); }

template<class T> inline T max(vector<T> a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a, int s, int n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<typename A, size_t N> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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> inline A min(A (&a)[N][O]) {

    A res = min(a[0]);

    rep(i, N)res = min(res, max(a[i]));

    return res;

}

template<typename A, size_t N, size_t O, size_t P> inline 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> inline 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> inline 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> inline 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> inline T sum(T &v) { return v; }

template<class T> inline auto sum(vector<T> &v) {

    auto ret = sum(v[0]);

    rep(i, 1, sz(v))ret += sum(v[i]);

    return ret;

}

template<class T> inline 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> inline 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, class U, class... V> inline auto sum(vector<T> &v, U head, V... tail) {

    auto ret = sum(v[0], tail...);

    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);

    return ret;

}



inline P sump(vp &v, int len = -1) {

    if (len == -1)len = v.size();

    P res = {0, 0};

    chmi(len, v.size());

    rep(i, len) {

        res.fi += v[i].fi;

        res.se += v[i].se;

    }

    return res;

}



///要素が0の時、返り値は0か1か

template<class T> inline T mul(vector<T> &v, int

len = -1

) {

    if (len == -1)

        len = v.size();

    T res = 1;

    chmi(len, v

            .

                    size()

    );

    rep(i, len)res *= v[i];

    return

            res;

}

inline void clear(PQ &q) {q=PQ();}

template<class T> inline void clear(queue<T> &q) { while (q.size())q.pop(); }

template<class T> inline T *negarr(int size) {

    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));

    return body + size;

}

template<class T> inline T *negarr2(int h, int w) {

    double **dummy1 = new double *[2 * h + 1];

    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];

    dummy1[0] = dummy2 + w;

    for (int 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> inline 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> inline vi imo(vector<T> &a, U

k) {

    vector<T> ret = a;

    rep(i, sz(ret) - 1)ret[i + 1] = ret[i] + (a[i] == k);

    return

            ret;

}

template<class T> inline 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> inline 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> struct ruiC {

    const vector<T> rui;

    ruiC(vector<T> &ru) : rui(ru) {}

    inline T operator()(int l, int r) {

        assert(l <= r);

        return rui[r] - rui[l];

    }

    inline T operator[](int i) {

        return rui[i];

    }

};



template<class T> struct rruic {

    const T *rrui;

    rruic(T *ru) : rrui(ru) {}

    //n-1から-1へ

    inline T operator()(int l, int r) {

        assert(l >= r);

        return rrui[r] - rrui[l];

    }

    inline T operator[](int i) { return rrui[i]; }

};

template<class T> inline 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> inline ruiC<T> ruic(vector<T> &a) {

    vector<T> ret = ruiv(a);

    return ruiC<T>(ret);

}

//kと同じものの数

template<class T, class U> inline 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> inline ruiC<int> ruic(T &a, U k) {

    vi ret = ruiv(a, k);

    return ruiC<int>(ret);

}



//xor

template<class T> inline 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> inline 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> inline vector<T> ruimi(vector<T> &a) {

    int 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> rruic<T> rrui(vector<T> &a) {

    int len = a.size();

    T *body = (T *) malloc((len + 1) * sizeof(T));

    T *res = body + 1;

    rer(i, len - 1)res[i - 1] = res[i] + a[i];

    return rruic<T>(res);

}

//掛け算

template<class T> T *rruim(vector<T> &a) {

    int 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(T &a, U v = 1) { a += v; }

template<class T, class U> void inc(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        inc(u, v

        );

}

template<class T> void inc(vector<T> &a) { for (auto &u :a)inc(u, 1); }

template<class T, class U> void plus(T &a, U v = 1) { a += v; }

template<class T, class U> void plus(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        inc(u, v

        );

}

template<class T> void plus(vector<T> &a) { for (auto &u :a)inc(u, 1); }

template<class T, class U> void dec(T &a, U v = 1) { a -= v; }

template<class T, class U> void dec(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        dec(u, v

        );

}

template<class T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }

template<class T, class U> void minu(T &a, U v = 1) { a -= v; }

template<class T, class U> void minu(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        dec(u, v

        );

}

template<class T> void minu(vector<T> &a) { for (auto &u :a)dec(u, 1); }

template<class U> void minu(string &a, U v = 1) { for (auto &u :a)dec(u, v); }

void minu(string &a) { for (auto &u :a)dec(u, 1); }

inline bool inside(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }

inline bool inside(int l, int v, int r) { return l <= v && v < r; }

#define ins inside

ll u(ll a) { return a < 0 ? 0 : a; }

template<class T> vector<T> u(const vector<T> &a) {

    vector<T> ret = a;

    fora(v, ret)v = u(v);

    return ret;

}

#define MIN(a) numeric_limits<a>::min()

#define MAX(a) numeric_limits<a>::max()

ll goldd(ll left, ll right, function<ll(ll)> 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;

}

ll goldt(ll left, ll right, function<ll(ll)> 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;

}

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;

}



constexpr bool bget(ll m, int keta) {

    assert(keta < 64);

    return (m >> keta) & 1;

}

int bget(ll m, int keta, int sinsuu) {

    m /= (ll) pow(sinsuu, keta);

    return m % sinsuu;

}

ll bit(int n) { return (1LL << (n)); }

ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }

int mask(int n) { return (1ll << n) - 1; }

//int bcou(ll m) { return __builtin_popcount(m & 0xFFFFFFFF) + __builtin_popcount(m >> 32); }

#define bcou __builtin_popcountll

//最下位ビット

int lbit(int n) {

    return n & -n;

}

//最上位ビット

int hbit(int n) {

    n |= (n >> 1);

    n |= (n >> 2);

    n |= (n >> 4);

    n |= (n >> 8);

    n |= (n >> 16);

    n |= (n >> 32);

    return n - (n >> 1);

}

int hbitk(int n) {

    int k = 0;

    rer(i, 5) {

        int a = k + (1ll << i);

        int b = 1ll << a;

        if (b <= n)k += 1ll << i;

    }

    return k;

}

//初期化は0を渡す

ll nextComb(ll &mask, int n, int 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

vl bitCombList(int n, int r) {

    vl res;

    int m = 0;

    while (nextComb(m, n, r)) {

        res.pb(m);

    }

    return res;

}

//大文字小文字を区別する

int altoiaZ(char c) {

    if ('A' <= c && c <= 'Z')return c - 'A';

    return c - 'a' + 26;

}

char itoalaZ(int i) {

    if (i < 26)return 'A' + i;

    return 'a' + i - 26;

}

//aもAも0を返す 基本小文字

int altoi(char c) {

    if ('A' <= c && c <= 'Z')return c - 'A';

    return c - 'a';

}

char itoal(int i) {

    return 'a' + i;

}

int ctoi(char c) { return c - '0'; }

char itoc(int i) { return i + '0'; }

int vtoi(vi &v) {

    int 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(int i) {

    vi res;

    while (i) {

        res.pb(i % 10);

        i /= 10;

    }

    rev(res);

    return res;

}

vi stov(string &a) {

    int n = sz(a);

    vi ret(n);

    rep(i, n) {

        ret[i] = a[i] - '0';

    }

    return ret;

}

vector<vector<int>> ctoi(vector<vector<char>> s, char c) {

    int n = sz(s), m = sz(s[0]);

    vector<vector<int>> res(n, vector<int>(m));

    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;

    return res;

}

#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );



//[i] := i番として圧縮されたものを返す

vi compress(vi &a) {

    vi b;

    int len = a.size();

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) {

        a[i] = lower_bound(all(b), a[i]) - b.begin();

    }

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

vi compress(vi &a, umap<int, int> &map) {

    vi b;

    int len = a.size();

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) {

        int v = a[i];

        a[i] = lower_bound(all(b), a[i]) - b.begin();

        map[v] = a[i];

    }

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}



vi compress(vi &a, vi &r) {

    vi b;

    int len = a.size();

    fora(v, a)b.pb(v);

    fora(v, r)b.pb(v);

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();

    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();

    int 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;

    int len = a.size();

    fora(v, a)b.pb(v);

    fora(v, r)b.pb(v);

    fora(v, s)b.pb(v);

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();

    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();

    for (int i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

vi compress(_vvi &a) {

    vi b;

    fora(vv, a)fora(v, vv)b.pb(v);

    sort(b);

    unique(b);

    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();

    int 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.pb(v);

    sort(b);

    unique(b);

    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

void compress(int a[], int len) {

    vi b;

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int 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 lowerBound(a, v) (*lower_bound(all(a),v))

#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())

#define upperBound(a, v) (*upper_bound(all(a),v))

#define ans(a) cout<<a<<endl;continue;

#define poll(a) q.front();q.pop()

#define dpoll(a) q.front();q.pop_front()

#define pollLast(a) q.back();q.pop_back()

#define pollBack(a) q.back();q.pop_back()

template<class T> inline void fin(T s) { cout << s << endl, exit(0); }



//便利 数学 math

int mod(int a, int m) { return (a % m + m) % m; }

int pow(int a) { return a * a; };

ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }

ll comi(int n, int 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 comd(int n, int r) {

    assert(n < 2020);

    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];

}

int gcd(int a, int b) {

    while (b) a %= b, swap(a, b);

    return abs(a);

}

int 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) {

    int 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;

    }

}

//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); }

ll sig(ll t) { return (1 + t) * t / 2; }

ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }





namespace pnumber_tools {

#define Eratosthenes30_2  class Eratosthenes {public:virtual void initialize() {} virtual void generate(const int64_t x) = 0;        virtual void generate(const int64_t from, const int64_t to) = 0;        virtual int64_t count() = 0;        virtual bool prime(int x) = 0;        virtual int32_t version() = 0;        inline uint64_t PopCnt(uint64_t);        inline int BitToIndex(uint8_t);    };    uint64_t Eratosthenes::PopCnt(uint64_t bits) {        bits = (bits & 0x5555555555555555ULL) + ((bits >> 1) & 0x5555555555555555ULL);        bits = (bits & 0x3333333333333333ULL) + ((bits >> 2) & 0x3333333333333333ULL);        bits = (bits & 0x0f0f0f0f0f0f0f0fULL) + ((bits >> 4) & 0x0f0f0f0f0f0f0f0fULL);        return (bits * 0x0101010101010101ULL) >> (64 - 8);    }    int Eratosthenes::BitToIndex(uint8_t b) {        switch (b) {            case 1 << 0:                return 0;            case 1 << 1:                return 1;            case 1 << 2:                return 2;            case 1 << 3:                return 3;            case 1 << 4:                return 4;            case 1 << 5:                return 5;            case 1 << 6:                return 6;            case 1 << 7:                return 7;        }        return -1;    }    namespace {        const uint64_t kSegmentSize = 1000000;          const int32_t kMod30[] = {1, 7, 11, 13, 17, 19, 23, 29};        const int C1[] = {6, 4, 2, 4, 2, 4, 6, 2};        const int C0[][8] = {                {0, 0, 0, 0, 0, 0, 0, 1},                {1, 1, 1, 0, 1, 1, 1, 1},                {2, 2, 0, 2, 0, 2, 2, 1},                {3, 1, 1, 2, 1, 1, 3, 1},                {3, 3, 1, 2, 1, 3, 3, 1},                {4, 2, 2, 2, 2, 2, 4, 1},                {5, 3, 1, 4, 1, 3, 5, 1},                {6, 4, 2, 4, 2, 4, 6, 1},        };        const uint8_t kMask[][8] = {                {0xfe, 0xfd, 0xfb, 0xf7, 0xef, 0xdf, 0xbf, 0x7f},                {0xfd, 0xdf, 0xef, 0xfe, 0x7f, 0xf7, 0xfb, 0xbf},                {0xfb, 0xef, 0xfe, 0xbf, 0xfd, 0x7f, 0xf7, 0xdf},                {0xf7, 0xfe, 0xbf, 0xdf, 0xfb, 0xfd, 0x7f, 0xef},                {0xef, 0x7f, 0xfd, 0xfb, 0xdf, 0xbf, 0xfe, 0xf7},                {0xdf, 0xf7, 0x7f, 0xfd, 0xbf, 0xfe, 0xef, 0xfb},                {0xbf, 0xfb, 0xf7, 0x7f, 0xfe, 0xef, 0xdf, 0xfd},                {0x7f, 0xbf, 0xdf, 0xef, 0xf7, 0xfb, 0xfd, 0xfe},        };        const int64_t kSievedPrimes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23};        const int64_t kInitialSize = 7 * 11 * 13 * 17 * 19 * 23;class Eratosthenes2 final : public Eratosthenes {    public:        void generate(const int64_t x) override;        void generate(const int64_t, const int64_t) override { assert(false); }        bool prime(int x) override;        int64_t count() override;        int32_t version() override { return 2; }        std::vector<uint8_t> flags_;    };    void Eratosthenes2::generate(const int64_t x) {        flags_.clear();        if (x > 10000000000)            return;        int64_t size = x / 30 + (x % 30 != 0);        flags_.resize(size, 0xff);        if (int r = x % 30) {            if (r <= 1) flags_[size - 1] = 0x0;            else if (r <= 7) flags_[size - 1] = 0x1;            else if (r <= 11) flags_[size - 1] = 0x3;            else if (r <= 13) flags_[size - 1] = 0x7;            else if (r <= 17) flags_[size - 1] = 0xf;            else if (r <= 19) flags_[size - 1] = 0x1f;            else if (r <= 23) flags_[size - 1] = 0x3f;            else if (r <= 29) flags_[size - 1] = 0x7f;        }        flags_[0] = 0xfe;        const uint64_t sqrt_x = std::ceil(std::sqrt(x) + 0.1);        const uint64_t sqrt_xi = sqrt_x / 30 + 1;        for (uint64_t i = 0; i < sqrt_xi; ++i) {            for (uint8_t flags = flags_[i]; flags; flags &= flags - 1) {                uint8_t lsb = flags & (-flags);                int ibit = BitToIndex(lsb);                const int32_t m = kMod30[ibit];                int32_t pm = 30 * i + 2 * m;                for (uint64_t j = i * pm + (m * m) / 30, k = ibit; j < flags_.size();                        j += i * C1[k] + C0[ibit][k], k = (k + 1) & 7) {                    flags_[j] &= kMask[ibit][k];                }            }        }    }int64_t Eratosthenes2::count() {            if (flags_.empty())                return -1;            int64_t ret = 3;            for (uint8_t f : flags_) {                ret += PopCnt(f);            }            return ret;        }        inline bool Eratosthenes2::prime(int x) {            static unordered_map<int, int> imap = {{1,  0},                                                   {7,  1},                                                   {11, 2},                                                   {13, 3},                                                   {17, 4},                                                   {19, 5},                                                   {23, 6},                                                   {29, 7}};            if (x == 2 || x == 3 || x == 5)return true;            else if (!(x & 1) || x % 3 == 0 || x % 5 == 0)return false;            return (flags_[x / 30] >> imap[x % 30]) & 1;        }}

    Eratosthenes30_2

    Eratosthenes2 era;

    int prime_len = 3e6;

    vector<int> primes;

    void setprime(int size = 3e6) {

        prime_len = size;

        era.generate(size);

        primes = {2, 3, 5};

        int nums[129] = {};

        nums[1] = 1;

        nums[2] = 7;

        nums[4] = 11;

        nums[8] = 13;

        nums[16] = 17;

        nums[32] = 19;

        nums[64] = 23;

        nums[128] = 29;

        //p = 30i + j

        int j = 0;

        int sflim = era.flags_.size();

        int i30 = 0;

        for (int i = 0; i < sflim; i++, i30 += 30) {

            uint8_t m = era.flags_[i];

            while (m) {

                j = m & -m;

                primes.push_back(i30 + nums[j]);

                m ^= j;

            }

        }

    }

    using u32 = unsigned int;//longlongになっていると思うが放置していいと思う

    using u64 = unsigned long long;

    using u128 = __uint128_t;



    //O(k log(n)^3)

    //32bit -> 270回程度

    //64bit -> 1260回程度

    //64bit以下で精度100%

    template<class BinOp> bool miller_rabin(const uint64_t &n, const uint64_t *witness, BinOp modmul) {

        if (n == 2) return true;

        if (n < 2 || n % 2 == 0) return false;

        const int64_t m = n - 1, d = m / (m & -m);

        auto modpow = [&](int64_t a, int64_t b) {

            int64_t res = 1;

            for (; b; b /= 2) {

                if (b & 1) res = modmul(res, a);

                a = modmul(a, a);

            }

            return res;

        };

        auto suspect = [&](uint64_t a, uint64_t t) {

            a = modpow(a, t);

            while (t != n - 1 && a != 1 && a != n - 1) {

                a = modmul(a, a);

                t = modmul(t, 2);

            }

            return a == n - 1 || t % 2 == 1;

        };

        for (const uint64_t *w = witness; *w; w++) {

            if (*w % n != 0 && !suspect(*w, d)) return false;

        }

        return true;

    }



//miller_rabin & エラトステネス

    bool prime(const int &n) {

        assert(n < 9223372036854775808ULL);

        if (!sz(primes))setprime();

        //エラトステネスの篩 偶数省略ver

        //[i] = i*2+1

        if (n < prime_len)return era.prime(n);

        if (n < 4294967296) {

            rep(i, 30) if (!(n % primes[i]))return false;

            constexpr uint64_t witness[] = {2, 7, 61, 0};

            auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return a * b % n; };

            return miller_rabin(n, witness, modmul);

        } else {

            rep(i, 50) if (!(n % primes[i]))return false;

            constexpr uint64_t witness[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022, 0};

            auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return (uint64_t) ((__uint128_t) a * b % n); };

            return miller_rabin(n, witness, modmul);

        }

    }

    vi divisors(int v) {

        vi res;

        double lim = std::sqrt(v);

        for (int i = 1; i <= lim; ++i) {

            if (v % i == 0) {

                res.pb(i);

//                if (i != v / i)res.pb(v / i);

            }

        }

        rer(i, sz(res) - 1) {

            if (res[i] != v / res[i])res.pb(v / res[i]);

        }

        return res;

    }

    vp factorization(int v) {

        assert(v);

        int tv = v;

        vp res;

        if (!sz(primes))setprime();

        for (auto &&p :primes) {

            if (v % p == 0)res.eb(p, 0);

            while (v % p == 0) {

                v /= p;

                res.back().second++;

            }

            if (v == 1 || p * p > tv)break;

        }

        if (v > 1)res.eb(v, 1);

        return res;

    }



    //注意 gcdを取る間隔 i%10 が場合によってはtleを引き起こす

    //うまくいかなかったら%1で動かすように変更した

    vp rho(int v) {

        assert(v);

        if (v == 1)return vp();

        vi ret;

        if (!sz(primes)) setprime();

        int lim = 300;                      //64bit

        if (v < 4294967296)lim = 80;      //32bit

        else if (v < 281474976710656LL)lim = 120; //48bit

        for (int i = 0; i < lim; i += 5) {

            //v<=1の判定回数を減らせる分高速化できる事を期待している

            while (v % primes[i] == 0) ret.pb(primes[i]), v /= primes[i];

            while (v % primes[i + 1] == 0) ret.pb(primes[i + 1]), v /= primes[i + 1];

            while (v % primes[i + 2] == 0) ret.pb(primes[i + 2]), v /= primes[i + 2];

            while (v % primes[i + 3] == 0) ret.pb(primes[i + 3]), v /= primes[i + 3];

            while (v % primes[i + 4] == 0) ret.pb(primes[i + 4]), v /= primes[i + 4];

            if (v <= 1) {

                sort(ret);//試し割りだけで終わった時

                return run_length(ret);

            }

        }

        int lower_prime_bound = primes[lim - 1] * primes[lim - 1];



        //試し割りするか

        mt19937 mr;

        mr.seed(clock());

        int c;

        auto fac = [&](int v, const int loop = 10) {

//        if (v == 4)return 2ll; //試し割りで消えた

            c = mr() % v;

            int x = mr() % v;

            int y = x;

            int d = 1;

            int lim = std::sqrt(v) + 2;

            //brentによる改良 差分を一操作で求められる

            for (int i = 1, mul = 1, next_2pow = 2; d == 1 && i <= lim; i++) {

                if (i == next_2pow) {

                    y = x;

                    next_2pow <<= 1;

                }

                x = ((u128) x * x + c) % v;

                mul = ((u128) mul * abs(x - y)) % v;//ユークリッドの互除法と同じ原理で因数は保たれる

                if (i % loop == 0) {//10とかなんとか iによっては死ぬ

                    d = gcd(mul, v);

                    mul = 1;

                }

            }

            if (d == v) return -1ll;

            return d;

        };



        queue<int> q;

        q.push(v);

        while (!q.empty()) {

            int v = q.front();

            q.pop();

            if (prime(v)) {

                ret.pb(v);

                continue;

            }

            int d = -1;

            //基本loop回数10でやり、駄目なら1で

            for (int i = 0; d == -1 && i < 5; i++)d = fac(v, 10);

            if (d == -1)while (d == -1)d = fac(v, 1);

            int d2 = v / d;

            rep(_, 2) {

                if (d <= lower_prime_bound)ret.pb(d);

                else q.push(d);

                d = d2;

            }

        }

        sort(ret);

        return run_length(ret);

    }

}

using namespace pnumber_tools;

//幾何 Pをcomplexとして扱う

template<class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }

dou atan2(pd a) { return atan2(a.se, a.fi); }

dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }

dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }

//bを中心とするabcのtheta aからcにかけて時計回り

dou angle(pd a, pd b, pd c) {

    dou ax = a.fi - b.fi;

    dou ay = a.se - b.se;

    dou cx = c.fi - b.fi;

    dou cy = c.se - b.se;

    double ret = atan2(cy, cx) - atan2(ay, ax);

    if (ret < 0) ret += 2 * PI;

    return ret;

}



dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }

dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }



template<class T> struct edge {

    int f, t;

    T c;

    int id;

    int type;

    edge(int f, int t, T c = 1, int id = -1, int ty = -1) : f(f), t(t), c(c), id(id), type(ty) {}

    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 {

protected:

    vector<bool> _used;

public :

    vector<vector<edge<T>>> g;

    vector<edge<T>> edges;

    int n;

    graph(int n) : n(n) { g.resize(n), _used.resize(n); }

    void clear() { g.clear(), edges.clear(); }

    void resize(int n) {

        this->n = n;

        g.resize(n);

        _used.resize(n);

    }

    int size() { return g.size(); }

    vector<edge<T> > &operator[](int i) { return g[i]; }

    virtual void add(int f, int t, T c, int id, int ty) = 0;

    virtual bool used(edge<T> &e) = 0;

    virtual bool used(int id) = 0;

    virtual void del(edge<T> &e) = 0;

    virtual void del(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;

    using graph<T>::_used;

    int eid = 0;



    digraph(int n) : graph<T>(n) {}

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("digraph add");

            deb(f, t, c, id, ty);

            ole();

        }

        if (id == -1)id = eid++;

        g[f].emplace_back(f, t, c, id, ty);

        edges.emplace_back(f, t, c, id, ty);

    }

    bool used(edge<T> &e) { return _used[e.id]; }

    bool used(int id) { return _used[id]; }

    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }

    void del(int id) { _used[id] = _used[id ^ 1] = 1; }

    void set_edges() {

        if (sz(edges))return;

        rep(i, n)fora(e, g[i])edges.pb(e);

    }

};

template<class T=int> class undigraph : public graph<T> {

public:

    using graph<T>::g;

    using graph<T>::n;

    using graph<T>::edges;

    using graph<T>::_used;

    int eid = 0;

    undigraph(int n) : graph<T>(n) {

    }

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("undigraph add");

            deb(f, t, c, id, ty);

            ole();

        }

        if (id == -1)id = eid, eid += 2;

        g[f].emplace_back(f, t, c, id, ty);

        g[t].emplace_back(t, f, c, id + 1, ty);

        edges.emplace_back(f, t, c, id, ty);

        edges.emplace_back(t, f, c, id + 1, ty);

    }

    void add(edge<T> &e) {

        int f = e.f, t = e.t, ty = e.type;

        T c = e.c;

        add(f, t, c, ty);

    }

    bool used(edge<T> &e) { return _used[e.id]; }

    bool used(int id) { return _used[id]; }

    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }

    void del(int id) { _used[id] = _used[id ^ 1] = 1; }

    void set_edges() {

        if (sz(edges))return;

        rep(i, n)fora(e, g[i])edges.pb(e);

    }

};

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; }

    inline int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }

    void push(u64 key, const T &value) {

        ++size;

        v[getbit(key ^ last)].emplace_back(key, value);

    }

    pair<u64, T> pop() {

        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;

    }

};

vi dijkstra2(const graph<int> &g, int s, int cant_arrive = -1) {

    if (!(0 <= s && s < g.n)) {

        debugline("dijkstra");

        deb(s, g.n);

        ole();

    }

    int initValue = MAX(int);

    vi dis(g.n, initValue);

    radixheap<int> q;

    dis[s] = 0;

    q.push(0, s);

    while (!q.empty()) {

        int nowc, i;

        tie(nowc, i) = q.pop();

        if (dis[i] != nowc)continue;

        for (auto &&e  : g.g[i]) {

            int to = e.t;

            int c = nowc + e.c;

            if (dis[to] > c) {

                dis[to] = c;

                q.push(dis[to], to);

            }

        }

    }

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;

    return dis;

}



template<class T> vector<T> dijkstra(const graph<T> &g, int s, int cant_arrive = -1) {

    if (!(0 <= s && s < g.n)) {

        debugline("dijkstra");

        deb(s, g.n);

        ole();

    }

    T initValue = MAX(T);

    vector<T> dis(g.n, initValue);

    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;

                q.emplace(dis[to], to);

            }

        }

    }

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;

    return dis;

}

template<class T> vector<vector<T>> warshall(const graph<T> &g, int cant_arrive = -1) {

    int n = g.n;

    vector<vector<T> > dis(n, vector<T>(n, linf));

    rep(i, n)fora(e, g.g[i])chmi(dis[e.f][e.t], e.c);

    rep(i, n)dis[i][i] = 0;

    rep(k, n)rep(i, n)rep(j, n)chmi(dis[i][j], dis[i][k] + dis[k][j]);

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)rep(i, n)rep(j, n) if (dis[i][j] == linf)dis[i][j] = -1;

    return dis;

}



template<class T=int> class tree : public undigraph<T> {

public:

    using undigraph<T>::g;

    using undigraph<T>::n;

    using undigraph<T>::edges;

    using undigraph<T>::_used;

    vi disv;

private:

    bool never = 1;

    int root = -1;

    vi dep;//到着した順の深さ

    vi vis;//到着した順

    vi first;//初めてiに到着した時

    vector<int> bestt;//タイミングを持つ

    int sn = 0;//seg木の底辺の長さ euler tourの長さ

    void built() {

        never = 0;

        n = g.size();

        //euler tour

        disv.resize(n);

        first.resize(n);

        dep.resize(n * 2);

        vis.resize(n * 2);

        function<void(int, int, int, int &)> dfs = [&](int i, int p, int d, int &k) {

            first[i] = k;

            vis[k] = i;

            dep[k++] = d;

            forg(gi, g[i]) {

                if (t == p)continue;

                disv[t] = disv[i] + c;

                dfs(t, i, d + 1, k);

                vis[k] = i;

                dep[k++] = d;

            }

        };

        int k = 0;

        dfs(root, -1, 0, k);

        //rmq

        sn = 1;

        while (sn < n * 2)sn <<= 1;



        bestt.resize((sn << 1) - 1);

        rep(i, k)bestt[i + sn - 1] = i;//時間を持つ

        rer(i, sn - 2) {

            int lt = (i << 1) + 1, rt = (i << 1) + 2;

            if (dep[bestt[lt]] < dep[bestt[rt]]) {

                bestt[i] = bestt[lt];

            } else {

                bestt[i] = bestt[rt];

            }

        }

    }

    //最小の深さを持つタイミングが知りたい

    inline int lca_rmq(int l, int r) {

        int midv = inf;

        int midi = -1;

        l += sn - 1;

        r += sn - 1;

        while (l < r) {

            //右なら、次に右上へ移動してしまうので

            if (!(l & 1)) {

                if (chmi(midv, dep[bestt[l]])) {

                    midi = bestt[l];

                }

            }

            //右なら、左を取らないと機会を逃す

            if (!(r & 1)) {

                if (chmi(midv, dep[bestt[r - 1]])) {

                    midi = bestt[r - 1];

                }

            }

            l >>= 1;

            r = (r - 1) >> 1;

        }

        return vis[midi];

    }



public:

    tree(int n, int root = 0) : undigraph<T>(n), root(root) {}

    bool leaf(int v) {

        return sz(g[v]) == 1 && v != root;

    }

    int lca(int u, int v) {

        if (never)built();

        if (first[u] > first[v])swap(u, v);

        return lca_rmq(first[u], first[v] + 1);

    }

    int dis(int u, int v) {

        if (never) built();

        if (first[u] > first[v])swap(u, v);

        int p = lca_rmq(first[u], first[v] + 1);

        return disv[u] + disv[v] - disv[p] * 2;

    }



};

//辺によりメモリを大量消費

// よってedgesを消している

//頂点10^6でメモリを190MB(制限の8割)使う



//軽量化のため、辺を持たないbig gridクラスがあってもいいかもしれない

//

template<class T=int> class grid_k6 : public undigraph<T> {

public:

    using undigraph<T>::g;

    using undigraph<T>::n;

    using undigraph<T>::edges;

    using undigraph<T>::_used;

    int H, W;

    int eid = 0;

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("grid_k6 add");

            deb(f, t, c, id, ty);

            ole();

        }

        g[f].emplace_back(f, t, c, eid++, ty);

        g[t].emplace_back(t, f, c, eid++, ty);

    }

    int getid(int h, int w) {

        if (!inside(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);

    }

    grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {

        rep(h, H) {

            rep(w, W) {

                int f = getid(h, w);

                if (w + 1 < W) add(f, getid(h, w + 1));

                if (h + 1 < H)add(f, getid(h + 1, w));

            }

        }

    }

    grid_k6(_vvc 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));

                }

            }

        }

    }

    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.pb(e);

    }

};

//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;

    using digraph<T>::_used;

    int H, W;

    int eid = 0;

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("digrid_k6 add");

            deb(f, t, c, id, ty);

            ole();

        }

        g[f].emplace_back(f, t, c, eid++, ty);

    }

    int getid(int h, int w) {

        if (!inside(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) {

        rep(h, H) {

            rep(w, W) {

                int f = getid(h, w);

                if (w + 1 < W) add(f, getid(h, w + 1));

                if (h + 1 < H)add(f, getid(h + 1, w));

            }

        }

    }

    digrid_k6(_vvc 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.pb(e);

    }

};

template<class T> bool nibu(const 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 1;

}

//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;



}

template<class T> ostream &operator<<(ostream &os, graph<T> &g) {

    rep(i, sz(g)) {

        forg(gi, g[i]) {

            os << f << " " << t << " " << c << endl;

        }

    }

    return os;

}



//機能拡張

vp vtop(vi &a, vi &b) {

    vp res(sz(a));

    rep(i, sz(a))res[i] = mp(a[i], b[i]);

    return res;

}

void ptov(vp &p, vi &a, vi &b) {

    a.resize(sz(p));

    b.resize(sz(p));

    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;

}

template<typename _CharT, typename _Traits, typename _Alloc>

basic_string<_CharT, _Traits, _Alloc>

operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {

    basic_string<_CharT, _Traits, _Alloc> __str(__lhs);

    __str.append(to_string(__rv));

    return __str;

}

template<typename _CharT, typename _Traits, typename _Alloc>

void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {

    __lhs += to_string(__rv);

}

template<typename _CharT, typename _Traits, typename _Alloc>

basic_string<_CharT, _Traits, _Alloc>

operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const signed __rv) {

    basic_string<_CharT, _Traits, _Alloc> __str(__lhs);

    __str.append(to_string(__rv));

    return __str;

}

template<typename _CharT, typename _Traits, typename _Alloc>

void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const signed __rv) {

    __lhs += to_string(__rv);

}

template<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, class U>

priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, vector <U> &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, class U> set<T> &operator+=(set<T> &a, U v) {

    a.insert(v);

    return a;

}

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.

            pb(v);

    return

            a;

}

template<class T, class U> vector<T> operator+(const vector <T> &a, U v) {

    vector<T> ret = a;

    ret += v;

    return ret;

}

template<class T, class U> vector<T> operator+(U v, const vector <T> &a) {

    vector<T> ret = a;

    ret.insert(ret.begin(), v);

    return ret;

}

template<class T> vector<T> operator+(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) {

    fora(v, b)a += v;

    return a;

}

template<class T, class U> vector<T> &operator+=(vector<T> &a, initializer_list<U>

v) {

    for (

        auto &&va

            :v)

        a.

                pb(va);

    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();

    }

    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<typename T> void remove(vector<T> &v, unsigned int

i) {

    v.

            erase(v

                          .

                                  begin()

                  + i);

}

template<typename T> void remove(vector<T> &v, unsigned int

s,

        unsigned int e

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + e);

}

template<typename T> void removen(vector<T> &v, unsigned int

s,

        unsigned int n

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + s + n);

}

template<typename T> void erase(vector<T> &v, unsigned int

i) {

    v.

            erase(v

                          .

                                  begin()

                  + i);

}

template<typename T> void erase(vector<T> &v, unsigned int

s,

        unsigned int e

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + e);

}

template<class T, class U> void erase(map<T, U> &m, int okl, int ngr) {

    m.erase(m.lower_bound(okl), m.lower_bound(ngr));

}

template<class T> void erase(set<T> &m, int okl, int ngr) {

    m.erase(m.lower_bound(okl), m.lower_bound(ngr));

}

template<typename T> void erasen(vector<T> &v, unsigned int

s,

        unsigned int n

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + s + n);

}

template<typename T, typename U> void insert(vector<T> &v, unsigned int

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 int

i,

        vector<T> list

) {

    for (

        auto &&va

            :list)

        v.

                insert(v

                               .

                                       begin()

                       + i++, va);

}

template<typename T, typename U> void insert(vector<T> &v, initializer_list<U>

list) {

    for (

        auto &&va

            :list)

        v.

                pb(va);

}

template<typename T, typename U> void insert(vector<T> &v, unsigned int

i,

        initializer_list<U> 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<typename T> void insert(set<T> &v, initializer_list<T> list) {

    for (auto &&va :list)v.insert(va);

}





//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.pb(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.pb(i);

        fora(e, g[i])if (--nyu[e.t] == 0)q.push(e.t);

    }

    return res.size() == n;

}

vector<string> split(const string a, const char deli) {

    string b = a + deli;

    int 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) {

    string b = a + deli;

    int l = 0, r = 0, n = b.size(), dn = deli.size();

    vector<string> res;

    rep(i, n) {

        if (i + dn <= n && b.substr(i, i + dn) == deli) {

            r = i;

            if (l < r)res.push_back(b.substr(l, r - l));

            i += dn - 1;

            l = i + 1;

        }

    }

    return res;

}

void yn(bool a) {

    if (a)cout << "yes" << endl;

    else cout << "no" << endl;

}

void Yn(bool a) {

    if (a)cout << "Yes" << endl;

    else cout << "No" << endl;

}

void YN(bool a) {

    if (a)cout << "YES" << endl;

    else cout << "NO" << endl;

}

void fyn(bool a) {

    if (a)cout << "yes" << endl;

    else cout << "no" << endl;

    exit(0);

}

void fYn(bool a) {

    if (a)cout << "Yes" << endl;

    else cout << "No" << endl;

    exit(0);

}

void fYN(bool a) {

    if (a)cout << "YES" << endl;

    else cout << "NO" << endl;

    exit(0);

}

void Possible(bool a) {

    if (a)cout << "Possible" << endl;

    else cout << "Impossible" << endl;

    exit(0);

}

void POSSIBLE(bool a) {

    if (a)cout << "POSSIBLE" << endl;

    else cout << "IMPOSSIBLE" << endl;

    exit(0);

}



int n, m, k, d, H, W, x, y, z, q;

int cou;

vi t, a, b, c;

//vvi (s, 0, 0);

vvc (ba, 0, 0);

vp p;

str s;



void solve() {

    cin >> n >> k;

    na(a, n);

    rep(300)

    {

        priority_queue<int, vector<int>, greater<int> > q;//小さい順

        rep(l, k + 1) {

            rep(r, k - l + 1) {

                if (l + r > n)break;

                int del = k - l - r;

                q = PQ();

                clear(q);

                int su = 0;

                rep(i, l) {

                    su += a[i];

                    q += a[i];

                }

                rep(i, r) {

                    su += a[n - 1 - i];

                    q += a[n - 1 - i];

                }

                rep(i, del) {

                    if (q.empty())break;

                    int d = q.top();

                    if (d >= 0)break;

                    q.pop();

                    su -= d;

                }

                chma(su);

            }

        }

    }

    cout << ma << endl;



}



int my(int n, vi &a) {

    return 0;

}

int sister(int n, vi &a) {

    int ret = 0;



    return ret;

}



signed main() {

    solve();



#define arg n,a

#ifdef _DEBUG

    bool bad = 0;

    for (int i = 0, ok = 1; i < k5 && ok; i++) {

        int n = rand(1, 8);

        vi a = ranv(n, 1, 10);

        int myres = my(arg);

        int res = sister(arg);

        ok = myres == res;

        if (!ok) {

            out(arg);

            cerr << "正解 : " << res << endl;

            cerr << "自分  : " << 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;

    }

#endif

    return 0;

};


// Faster C++ solution for the same problem (measured speedup: 1.4186506893809734x):
//todo 文字数を少なくする

//mintより高速に計算したい時用に modmul() modadd (正の値限定) を作る

//#pragma GCC optimize ("-O3")

#include <bits/stdc++.h>

using namespace std;

//@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));

    };

} __initon;

//衝突対策

#define ws ___ws



//@必須構造

struct T {

    int f, s, t;

    T() { f = -1, s = -1, t = -1; }

    T(int f, int s, int t) : f(f), s(s), t(t) {}

    bool operator<(const T &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; 大きい順

    }

    bool operator>(const T &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 T &r) const {

        return f == r.f && s == r.s && t == r.t;

    }

    bool operator!=(const T &r) const {

        return f != r.f || s != r.s || t != r.t;

    }

    int operator[](int i) {

        assert(i < 3);

        return i == 0 ? f : i == 1 ? s : t;

    }

};

struct F {

    int a, b, c, d;

    F() { a = -1, b = -1, c = -1, d = -1; }

    F(int a, int b, int c, int d) : a(a), b(b), c(c), d(d) {}

    bool operator<(const F &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 F &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 F &r) const {

        return a == r.a && b == r.b && c == r.c && d == r.d;

    }

    bool operator!=(const F &r) const {

        return a != r.a || b != r.b || c != r.c || d != r.d;

    }

    int operator[](int i) {

        assert(i < 4);

        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;

    }

};



T mt(int a, int b, int c) {

    return T(a, b, c);

}

//@マクロ省略系 型,構造



#define int long long

#define ll long long

#define double long double

#define ull unsigned long long

using dou = double;

using itn = int;

using str = string;

using bo= bool;

#define au auto

using P = pair<ll, ll>;

using pd =pair<dou, dou>;

#define fi first

#define se second

#define vec vector

#define beg begin

#define rbeg rbegin

#define con continue

#define bre break

#define brk break

#define is ==



#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))



//マクロ省略系 コンテナ



using vi = vector<int>;

#define _overloadvvi(_1, _2, _3, _4, name, ...) name

#define vvi0() vec<vi>

#define vvi1(a) vec<vi> a

#define vvi2(a, b) vec<vi> a(b)

#define vvi3(a, b, c) vec<vi> a(b,vi(c))

#define vvi4(a, b, c, d) vec<vi> a(b,vi(c,d))

#define vvi(...) _overloadvvi(__VA_ARGS__,vvi4,vvi3,vvi2 ,vvi1,vvi0)(__VA_ARGS__)





using vl = vector<ll>;

#define _overloadvvl(_1, _2, _3, _4, name, ...) name

#define vvl1(a) vec<vl> a

#define vvl2(a, b) vec<vl> a(b)

#define vvl3(a, b, c) vec<vl> a(b,vl(c))

#define vvl4(a, b, c, d) vec<vl> a(b,vl(c,d))

#define vvl(...) _overloadvvl(__VA_ARGS__,vvl4,vvl3,vvl2 ,vvl1)(__VA_ARGS__)



using vb = vector<bool>;

#define _overloadvvb(_1, _2, _3, _4, name, ...) name

#define vvb1(a) vec<vb> a

#define vvb2(a, b) vec<vb> a(b)

#define vvb3(a, b, c) vec<vb> a(b,vb(c))

#define vvb4(a, b, c, d) vec<vb> a(b,vb(c,d))

#define vvb(...) _overloadvvb(__VA_ARGS__,vvb4,vvb3,vvb2 ,vvb1)(__VA_ARGS__)



using vs = vector<string>;

#define _overloadvvs(_1, _2, _3, _4, name, ...) name

#define vvs1(a) vec<vs> a

#define vvs2(a, b) vec<vs> a(b)

#define vvs3(a, b, c) vec<vs> a(b,vs(c))

#define vvs4(a, b, c, d) vec<vs> a(b,vs(c,d))

#define vvs(...) _overloadvvs(__VA_ARGS__,vvs4,vvs3,vvs2 ,vvs1)(__VA_ARGS__)



using vd = vector<double>;

#define _overloadvvd(_1, _2, _3, _4, name, ...) name

#define vvd1(a) vec<vd> a

#define vvd2(a, b) vec<vd> a(b)

#define vvd3(a, b, c) vec<vd> a(b,vd(c))

#define vvd4(a, b, c, d) vec<vd> a(b,vd(c,d))

#define vvd(...) _overloadvvd(__VA_ARGS__,vvd4,vvd3,vvd2 ,vvd1)(__VA_ARGS__)



using vc=vector<char>;

#define _overloadvvc(_1, _2, _3, _4, name, ...) name

#define vvc1(a) vec<vc> a

#define vvc2(a, b) vec<vc> a(b)

#define vvc3(a, b, c) vec<vc> a(b,vc(c))

#define vvc4(a, b, c, d) vec<vc> a(b,vc(c,d))

#define vvc(...) _overloadvvc(__VA_ARGS__,vvc4,vvc3,vvc2 ,vvc1)(__VA_ARGS__)



using vp = vector<P>;

#define _overloadvvp(_1, _2, _3, _4, name, ...) name

#define vvp1(a) vec<vp> a

#define vvp2(a, b) vec<vp> a(b)

#define vvp3(a, b, c) vec<vp> a(b,vp(c))

#define vvp4(a, b, c, d) vec<vp> a(b,vp(c,d))

#define vvp(...) _overloadvvp(__VA_ARGS__,vvp4,vvp3,vvp2 ,vvp1)(__VA_ARGS__)



using vt = vector<T>;

#define _overloadvvt(_1, _2, _3, _4, name, ...) name

#define vvt1(a) vec<vt> a

#define vvt2(a, b) vec<vt> a(b)

#define vvt3(a, b, c) vec<vt> a(b,vt(c))

#define vvt4(a, b, c, d) vec<vt> a(b,vt(c,d))

#define vvt(...) _overloadvvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1)(__VA_ARGS__)



#define v3i(a, b, c, d) vector<vector<vi>> a(b, vector<vi>(c, vi(d)))

#define v3d(a, b, c, d) vector<vector<vd>> a(b, vector<vd>(c, vd(d)))

#define v3m(a, b, c, d) vector<vector<vm>> a(b, vector<vm>(c, vm(d)))



#define _vvi vector<vi>

#define _vvl vector<vl>

#define _vvb vector<vb>

#define _vvs vector<vs>

#define _vvd vector<vd>

#define _vvc vector<vc>

#define _vvp vector<vp>



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<int>(__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 PQ priority_queue<ll, vector<ll>, greater<ll> >

#define tos to_string

using mapi = map<int, int>;

using mapp = map<P, int>;

using mapd = map<dou, int>;

using mapc = map<char, int>;

using maps = map<str, int>;

using seti = set<int>;

using setd = set<dou>;

using setc = set<char>;

using sets = set<str>;

using qui = queue<int>;

#define bset bitset

#define uset unordered_set

#define mset multiset

#define umap unordered_map

#define umapi unordered_map<int,int>

#define umapp unordered_map<P,int>

#define mmap multimap



//マクロ 繰り返し

#define _overloadrep(_1, _2, _3, _4, name, ...) name

# define rep1(n) for(int rep1i = 0,_lim=n; rep1i < _lim ; rep1i++)

# define _rep(i, n) for(int i = 0,_lim=n; i < _lim ; i++)

#define repi(i, m, n) for(int i = m,_lim=n; i < _lim ; i++)

#define repadd(i, m, n, ad) for(int i = m,_lim=n; i < _lim ; i+= ad)

#define rep(...) _overloadrep(__VA_ARGS__,repadd,repi,_rep,rep1)(__VA_ARGS__)

#define _rer(i, n) for(int i = n; i >= 0 ; i--)

#define reri(i, m, n) for(int i = m,_lim=n; i >= _lim ; i--)

#define rerdec(i, m, n, dec) for(int i = m,_lim=n; i >= _lim ; i-=dec)

#define rer(...) _overloadrep(__VA_ARGS__,rerdec,reri,_rer,)(__VA_ARGS__)



#define reps2(i, j, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)

#define reps3(i, j, k, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)

#define reps4(i, j, k, l, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)for(int l = 0; l < _lim ; l++)



#define _overloadreps(_1, _2, _3, _4, _5, name, ...) name

#define reps(...) _overloadreps(__VA_ARGS__,reps4,reps3,reps2,_rep,)(__VA_ARGS__)



#define repss(i, j, k, a, b, c) for(int i = 0; i < a ; i++)for(int j = 0; j < b ; j++)for(int k = 0; k < c ; k++)



#define fora(a, b) for(auto&& a : b)

#define forg(gi, ve) for (int gi = 0,_lim = ve.size(), f, t, c; gi < _lim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)

#define fort(gi, ve) for (int gi = 0, f, t, c; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)if(t!=p)

//#define fort(gi, ve) for (int gi = 0, f, t, c;gi<ve.size()&& (gi+= (ve[gi].t==p))< ve.size() && (f = ve[gi].f,t=ve[gi].t, c = ve[gi].c,true); gi++)



#define form(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it).fi < r; it++)

#define forit(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r; it++)



//マクロ 定数

#define k3 1010

#define k4 10101

#define k5 101010

#define k6 1010101

#define k7 10101010

const int inf = (int) 1e9 + 100;

const ll linf = (ll) 1e18 + 100;

const char infc = '{';

const string infs = "{";

const double eps = 1e-9;

const double PI = 3.1415926535897932384626433832795029L;

ll ma = numeric_limits<ll>::min();

ll mi = numeric_limits<ll>::max();

const int y4[] = {-1, 1, 0, 0};

const int x4[] = {0, 0, -1, 1};

const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};

const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};



//マクロ省略形 関数等

#define arsz(a) (sizeof(a)/sizeof(a[0]))

#define sz(a) ((int)(a).size())

//#define rs resize

#define mp make_pair

#define pb push_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(int a) { return !(a & 1); }

constexpr bool od(int 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);

        }

    };

}

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 << endl;

    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 (int 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 (int i = 0; i < vec.size(); i++) {

        for (int j = 0; j < vec[0].size(); j++) {

            os << vec[i][j];

        }

        os << endl;

    }

    return os;

}

template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {

    for (auto &&v:m) os << v;

    return os;

}



template<typename V, typename H> void resize(vector<V> &vec,

        const H head

) {

    vec.

            resize(head);

}

template<typename V, typename H, typename ... T> void resize(vector<V> &vec,

        const H &head,

        const T ... tail

) {

    vec.

            resize(head);

    for (

        auto &v

            : vec)

        resize(v, tail

                ...);

}

template<typename T, typename _Pr> bool all_of(const vector<T> &vec, _Pr pred) { return std::all_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> bool any_of(const vector<T> &vec, _Pr pred) { return std::any_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> bool none_of(const vector<T> &vec, _Pr pred) { return std::none_of(vec.begin(), vec.end(), pred); }

template<typename T, typename _Pr> const typename vector<T>::const_iterator find_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred); }

template<class T> bool contains(const std::string &s, const T &v) { return s.find(v) != std::string::npos; }

template<typename T> bool contains(const vector<T> &vec, const T &val) { return std::find(vec.begin(), vec.end(), val) != vec.end(); }

template<typename T, typename _Pr> bool contains_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred) != vec.end(); }

template<class T, class U> void replace(vector<T> &a, T

key,

        U v

) {

    replace(a

                    .

                            begin(), a

                    .

                            end(), key, v

    );

}

void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }

void replace(str &a, str key, str v) {

    if (v == "")a.erase(remove(all(a), key[0]), a.end());

    else replace(all(a), key[0], v[0]);

}

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, '#'); }



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)); }



template<class T> T pop(set<T> &set) {

    T res = *set.begin();

    set.erase(set.find(res));

    return res;

}

template<class T> T pop(mset<T> &set) {

    T res = *set.begin();

    set.erase(set.find(res));

    return res;

}

template<class T> T popBack(set<T> &set) {

    T res = *set.rbegin();

    set.erase(set.find(res));

    return res;

}

template<class T> T popBack(mset<T> &set) {

    T res = *set.rbegin();

    set.erase(set.find(res));

    return res;

}

inline void sort(string &a) { sort(a.begin(), a.end()); }

template<class T> inline void sort(vector<T> &a) { sort(a.begin(), a.end()); }

template<class T> inline void sort(vector<T> &a, int

len) {

    sort(a

                 .

                         begin(), a

                                          .

                                                  begin()

                                  + len);

}

template<class T, class F> inline void sort(vector<T> &a, F

f) {

    sort(a

                 .

                         begin(), a

                 .

                         end(),

         [&](

                 T a, T

         b) {

             return

                     f(a)

                     <

                     f(b);

         });

}

enum ___pcomparator {

    fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd

};

inline void sort(vector<P> &a, ___pcomparator

type) {

    switch (type) {

        case fisi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi < r.fi : l.se < r.

                             se;

                 });

            break;

        case fisd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi < r.fi : l.se > r.

                             se;

                 });

            break;

        case fdsi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi > r.fi : l.se < r.

                             se;

                 });

            break;

        case fdsd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.fi != r.fi ? l.fi > r.fi : l.se > r.

                             se;

                 });

            break;

        case sifi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se < r.se : l.fi < r.

                             fi;

                 });

            break;

        case sifd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se < r.se : l.fi > r.

                             fi;

                 });

            break;

        case sdfi:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se > r.se : l.fi < r.

                             fi;

                 });

            break;

        case sdfd:

            sort(all(a),

                 [&](

                         P l, P

                 r) {

                     return l.se != r.se ? l.se > r.se : l.fi > r.

                             fi;

                 });

            break;

    }

};

inline void sort(vector<T> &a, ___pcomparator

type) {

    switch (type) {

        case fisi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f < r.f : l.s < r.

                             s;

                 });

            break;

        case

            fisd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f < r.f : l.s > r.

                             s;

                 });

            break;

        case

            fdsi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f > r.f : l.s < r.

                             s;

                 });

            break;

        case

            fdsd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.f != r.f ? l.f > r.f : l.s > r.

                             s;

                 });

            break;

        case

            sifi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s < r.s : l.f < r.

                             f;

                 });

            break;

        case

            sifd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s < r.s : l.f > r.

                             f;

                 });

            break;

        case

            sdfi:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s > r.s : l.f < r.

                             f;

                 });

            break;

        case

            sdfd:

            sort(all(a),

                 [&](

                         T l, T

                 r) {

                     return l.s != r.s ? l.s > r.s : l.f > r.

                             f;

                 });

            break;

    }

};

template<class T> inline void rsort(vector<T> &a) { sort(a.begin(), a.end(), greater<T>()); };

template<class T> inline void rsort(vector<T> &a, int

len) {

    sort(a

                 .

                         begin(), a

                                          .

                                                  begin()

                                  + len,

         greater<T>()

    );

};

template<class U, class F> inline void rsort(vector<U> &a, F

f) {

    sort(a

                 .

                         begin(), a

                 .

                         end(),

         [&](

                 U l, U

         r) {

             return

                     f(l)

                     >

                     f(r);

         });

};

template<class U> inline void sortp(vector<U> &a, vector<U> &b) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.eb(a[i], b[i]);

    sort(c);

    rep(i, n) {

        a[i] = c[i].first;

        b[i] = c[i].second;;

    }

};

//F = T<T>

//例えばreturn p.fi + p.se;

template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, F

f) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    sort(c, f

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, char

type) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    sort(c, type

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U> inline void rsortp(vector<U> &a, vector<U> &b) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.eb(a[i], b[i]);

    rsort(c);

    rep(i, n) {

        a[i] = c[i].first;

        b[i] = c[i].second;

    }

};

template<class U, class F> inline void rsortp(vector<U> &a, vector<U> &b, F

f) {

    vp c;

    int n = sz(a);

    assert(n == sz(b));

    rep(i, n)c.

                eb(a[i], b[i]

        );

    rsort(c, f

    );

    rep(i, n) {

        a[i] = c[i].

                first;

        b[i] = c[i].

                second;

    }

};

template<class U> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.eb(a[i], b[i], c[i]);

    sort(r);

    rep(i, n) {

        a[i] = r[i].f;

        b[i] = r[i].s;

        c[i] = r[i].t;

    }

};

template<class U, class F> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c, F

f) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.

                eb(a[i], b[i], c[i]

        );

    sort(r, f

    );

    rep(i, n) {

        a[i] = r[i].

                f;

        b[i] = r[i].

                s;

        c[i] = r[i].

                t;

    }

};

template<class U, class F> inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c, F

f) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.

                eb(a[i], b[i], c[i]

        );

    rsort(r, f

    );

    rep(i, n) {

        a[i] = r[i].

                f;

        b[i] = r[i].

                s;

        c[i] = r[i].

                t;

    }

};

template<class U> inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c) {

    vt r;

    int n = sz(a);

    assert(n == sz(b));

    assert(n == sz(c));

    rep(i, n)r.eb(a[i], b[i], c[i]);

    rsort(r);

    rep(i, n) {

        a[i] = r[i].f;

        b[i] = r[i].s;

        c[i] = r[i].t;

    }

};



//indexをつけてsortする (second)

template<class T> auto sorti(vector<T> &a) {

    int n = sz(a);

    vp ret(n);

    rep(i, n)ret[i] = P(a[i], i);

    sort(ret);

    return ret;

};

template<class T> auto rsorti(vector<T> &a) {

    int n = sz(a);

    vp ret(n);

    rep(i, n)ret[i] = P(a[i], i);

    rsort(ret);

    return ret;

};

template<class T> inline void sort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; i++)sort(a[i]); }

template<class T> inline void rsort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; i++)rsort(a[i]); }

template<typename A, size_t N, typename T> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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 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;

}



template<typename V, typename T>

void fill(V &xx, const T vall) {

    xx = vall;

}

template<typename V, typename T>

void fill(vector<V> &vecc,

        const T vall

) {

    for (

        auto &&vx

            : vecc)

        fill(vx, vall

        );

}



//@汎用便利関数 入力

inline int in() {

    int ret;

//    scanf("%lld", &ret);

    cin >> ret;

    return ret;

}

inline string sin() {

    string ret;

    cin >> ret;

    return ret;

}

template<class T> inline void in(T &head) { cin >> head; }

template<class T, class... U> inline void in(T &head, U &... tail) {

    cin >> head;

    in(tail...);

}



#define _overloaddin(_1, _2, _3, _4, name, ...) name

#define din1(a) int a;cin>>a

#define din2(a, b) int a,b;cin>>a>> b

#define din3(a, b, c) int a,b,c;cin>>a>>b>>c

#define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d

#define din(...) _overloaddin(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)



#define _overloaddind(_1, _2, _3, _4, name, ...) name

#define din1d(a) din1(a);a--

#define din2d(a, b) din2(a,b);a--,b--

#define din3d(a, b, c) din3(a,b,c);a--,b--,c--

#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--

#define dind(...) _overloaddind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)





#define _overloadout(_1, _2, _3, _4, name, ...) name

#define out1(a) cout<<a<<endl

#define out2(a, b) cout<<a<<" "<< b<<endl

#define out3(a, b, c) cout<<a<<" "<<b<<" "<<c<<endl

#define out4(a, b, c, d) cout<<a<<" "<<b<<" "<<c<<" "<<d<<endl

#define out(...) _overloadout((__VA_ARGS__),out4,out3,out2,out1)((__VA_ARGS__))



template<class T> void outl(vector<T> &a) { fora(v, a)cout << v << endl; }

#define na(a, n) a.resize(n); rep(_i,n) cin >> a[_i];

#define dna(a, n) vi a(n); rep(_i,n) cin >> a[_i];

#define nao(a, n) a.resize(n+1); rep(_i,n) cin >> a[_i+1];

#define nad(a, n) a.resize(n); rep(_i,n){ cin >> a[_i]; a[_i]--;}

#define na2(a, b, n) a.resize(n),b.resize(n);rep(_i, n)cin >> a[_i] >> b[_i];

#define dna2(a, b, n) vi a(n),b(n);rep(_i, n)cin >> a[_i] >> b[_i];

#define nao2(a, b, n) a.resize(n+1),b.resize(n+1);rep(_i, n)cin >> a[_i+1] >> b[_i+1];

#define na2d(a, b, n) a.resize(n),b.resize(n);rep(_i, n){cin >> a[_i] >> b[_i];a[_i]--,b[_i]--;}

#define na3(a, b, c, 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(_i, n)cin >> a[_i] >> b[_i] >> c[_i];

#define na3d(a, b, c, 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(_i, n){cin >> a[_i] >> b[_i] >> c[_i];a[_i]--,b[_i]--,c[_i]--;}

#define nt(a, h, w) resize(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi];

#define ntd(a, h, w) rs(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi], a[_hi][_wi]--;

#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(_hi,1,h+1)rep(_wi,1,w+1) cin >> a[_hi][_wi];



//デバッグ

#define sp << " " <<



#define debugName(VariableName) # VariableName



#define _deb1(x)  debugName(x)<<" = "<<x

#define _deb2(x, ...) _deb1(x) <<", "<< _deb1(__VA_ARGS__)

#define _deb3(x, ...) _deb1(x) <<", "<< _deb2(__VA_ARGS__)

#define _deb4(x, ...) _deb1(x) <<", "<< _deb3(__VA_ARGS__)

#define _deb5(x, ...) _deb1(x) <<", "<< _deb4(__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__)



#define _overloadebug(_1, _2, _3, _4, _5, _6, _7, _8, _9, _10, name, ...) name



#ifdef _DEBUG

#define deb(...)  cerr<< _overloadebug(__VA_ARGS__,_deb10,_deb9,_deb8,_deb7,_deb6,_deb5,_deb4,_deb3,_deb2,_deb1)(__VA_ARGS__) <<endl

#else

#define deb(...) ;

#endif



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



//よく使うクラス、構造体

class UnionFind {

public:

    vi par, rank, sizes;

    int n, trees;

    UnionFind(int n) : n(n), trees(n) {

        par.resize(n), rank.resize(n), sizes.resize(n);

        rep(i, n)par[i] = i, sizes[i] = 1;

    }

    int root(int x) {

        if (par[x] == x)return x;

        else return par[x] = root(par[x]);

    }

    int find(int x) { return root(x); }

    void unite(int x, int y) {

        x = root(x);

        y = root(y);

        if (x == y)return;

        if (rank[x] < rank[y])swap(x, y);

        trees--;

        par[y] = x;

        sizes[x] += sizes[y];

        if (rank[x] == rank[y])rank[x]++;

    }

    bool same(int x, int y) { return root(x) == root(y); }

    int size(int x) { return sizes[root(x)]; }

    //順不同 umapなので

    vec<vi> sets() {

        vec<vi> res(trees);

        umap<int, vi> map;

        rep(i, n) map[root(i)].push_back(i);

        int i = 0;

        for (auto &&p:map) {

            int r = p.fi;

            res[i].push_back(r);

            for (auto &&v:p.se) {

                if (r == v)continue;

                res[i].push_back(v);

            }

            i++;

        }

        return res;

    }

};



using bint =__int128;

using u32 = unsigned int;

using u64 = unsigned long long;

using u128 = __uint128_t;

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 = '-';

        }

        int len = std::end(buffer) - d;

        if (dest.rdbuf()->sputn(d, len) != len) {

            dest.setstate(std::ios_base::badbit);

        }

    }

    return dest;

}



__int128 toi128(string &s) {

    __int128 ret = 0;

    for (int i = 0; i < s.length(); i++)

        if ('0' <= s[i] && s[i] <= '9')

            ret = 10 * ret + s[i] - '0';

    return ret;

}





template<typename T> T minv(T a, T m);

template<typename T> T minv(T a);



template<typename T>

class Modular {

public:

    using Type = typename decay<decltype(T::value)>::type;



    constexpr Modular() : value() {}

    template<typename U>

    Modular(const U &x) {

        value = normalize(x);

    }



    template<typename U>

    static Type normalize(const U &x) {

        Type v;

        if (-mod() <= x && x < mod()) v = static_cast<Type>(x);

        else v = static_cast<Type>(x % mod());

        if (v < 0) v += mod();

        return v;

    }



    const Type &operator()() const { return value; }

    template<typename U>

    explicit operator U() const { return static_cast<U>(value); }

    constexpr static Type mod() { return T::value; }



    Modular &operator+=(const Modular &other) {

        if ((value += other.value) >= mod()) value -= mod();

        return *this;

    }

    Modular &operator-=(const Modular &other) {

        if ((value -= other.value) < 0) value += mod();

        return *this;

    }

    template<typename U> Modular &operator+=(const U &other) { return *this += Modular(other); }

    template<typename U> Modular &operator-=(const U &other) { return *this -= Modular(other); }

    Modular &operator++() { return *this += 1; }

    Modular &operator--() { return *this -= 1; }

    Modular operator++(signed) {

        Modular result(*this);

        *this += 1;

        return result;

    }

    Modular operator--(signed) {

        Modular result(*this);

        *this -= 1;

        return result;

    }

    Modular operator-() const { return Modular(-value); }



    template<typename U = T>

    typename enable_if<is_same<typename Modular<U>::Type, signed>::value, Modular>::type &operator*=(const Modular &rhs) {

#ifdef _WIN32

        uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);

        uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;

        asm(

        "divl %4; \n\t"

        : "=a" (d), "=d" (m)

        : "d" (xh), "a" (xl), "r" (mod())

        );

        value = m;

#else

        value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));

#endif

        return *this;

    }

    template<typename U = T>

    typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type &operator*=(const Modular &rhs) {

        int64_t q = static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());

        value = normalize(value * rhs.value - q * mod());

        return *this;

    }

    template<typename U = T>

    typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type &operator*=(const Modular &rhs) {

        value = normalize(value * rhs.value);

        return *this;

    }

    Modular &operator/=(const Modular &other) { return *this *= Modular(minv(other.value)); }

    template<typename U> friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);

    template<typename U> friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);

    template<typename U> friend std::istream &operator>>(std::istream &stream, Modular<U> &number);

    operator int() { return normalize(value); }

private:

    Type value;

};



template<typename T> bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value == rhs.value; }

template<typename T, typename U> bool operator==(const Modular<T> &lhs, U rhs) { return lhs == Modular<T>(rhs); }

template<typename T, typename U> bool operator==(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) == rhs; }



template<typename T> bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) { return !(lhs == rhs); }

template<typename T, typename U> bool operator!=(const Modular<T> &lhs, U rhs) { return !(lhs == rhs); }

template<typename T, typename U> bool operator!=(U lhs, const Modular<T> &rhs) { return !(lhs == rhs); }



template<typename T> bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value < rhs.value; }



template<typename T> Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }

template<typename T, typename U> Modular<T> operator+(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) += rhs; }

template<typename T, typename U> Modular<T> operator+(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }



template<typename T> Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }

template<typename T, typename U> Modular<T> operator-(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) -= rhs; }

template<typename T, typename U> Modular<T> operator-(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }



template<typename T> Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }

template<typename T, typename U> Modular<T> operator*(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) *= rhs; }

template<typename T, typename U> Modular<T> operator*(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }



template<typename T> Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U> Modular<T> operator/(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U> Modular<T> operator/(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }



constexpr signed MOD = 1e9 + 7;//MOD

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



constexpr int mint_len = 1400001;

vi fac, finv, inv;

vi p2;

void setmod() {

//    p2.resize(mint_len);p[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i] = p2[i - 1] * 2 % MOD;

    fac.resize(mint_len);

    finv.resize(mint_len);

    inv.resize(mint_len);

    inv[1] = fac[0] = finv[0] = 1;

    for (int i = 2; i < mint_len; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;

    for (int i = 1; i < mint_len; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;

}

#define smod setmod

inline mint com(int n, int r) {

    if (r < 0 || r > n) return 0;

    return mint(finv[r] * fac[n] % MOD * finv[n - r]);

}

inline mint pom(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return mint(fac[n] * finv[n - 1]);

}

inline mint npr(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return mint(fac[n] * finv[n - r]);

}

inline int nprin(int n, int r) {

//    if (!sz(fac)) com(0, -1);

    if (r < 0 || r > n) return 0;

    return fac[n] * finv[n - r] % MOD;

}



inline int icom(int n, int r) {

    const int NUM_ = 1400001;

    static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];

    if (fac[0] == 0) {

        inv[1] = fac[0] = finv[0] = 1;

        for (int i = 2; i <= NUM_; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;

        for (int i = 1; i <= NUM_; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;

    }

    if (r < 0 || r > n) return 0;

    return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;

}

#define ncr com

#define ncri icom

inline mint nhr(int n, int r) { return com(n + r - 1, r); }

inline mint hom(int n, int r) { return com(n + r - 1, r); }

inline int nhri(int n, int r) { return icom(n + r - 1, r); }





template<typename T> inline T minv(T a, T m) {

    T u = 0, v = 1;

    while (a != 0) {

        T t = m / a;

        m -= t * a;

        swap(a, m);

        u -= t * v;

        swap(u, v);

    }

    assert(m == 1);

    return u;

}

template<typename T> inline T minv(T a) {

    if (a < mint_len)return inv[a];

    T u = 0, v = 1;

    T m = MOD;

    while (a != 0) {

        T t = m / a;

        m -= t * a;

        swap(a, m);

        u -= t * v;

        swap(u, v);

    }

    assert(m == 1);

    return u;

}



template<typename T, typename U> inline Modular<T> mpow(const Modular<T> &a, const U &b) {

    assert(b >= 0);

    int x = a(), res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= MOD;

        (x *= x) %= MOD;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U, typename V> inline mint mpow(const T a, const U b, const V m = MOD) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= m;

        (x *= x) %= m;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U> inline mint mpow(const T a, const U b) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= MOD;

        (x *= x) %= MOD;

        p >>= 1;

    }

    return res;

}

template<typename T, typename U, typename V> inline int mpowi(const T &a, const U &b, const V &m = MOD) {

    assert(b >= 0);

    int x = a, res = 1;

    U p = b;

    while (p > 0) {

        if (p & 1) (res *= x) %= m;

        (x *= x) %= m;

        p >>= 1;

    }

    return res;

}

template<typename T> inline string to_string(const Modular<T> &number) {

    return to_string(number());

}



template<typename T> std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {

    return stream << number();

}



template<typename T> std::istream &operator>>(std::istream &stream, Modular<T> &number) {

    typename common_type<typename Modular<T>::Type, int64_t>::type x;

    stream >> x;

    number.value = Modular<T>::normalize(x);

    return stream;

}



using PM = pair<mint, mint>;

using vm = vector<mint>;

using mapm = map<int, mint>;

using umapm = umap<int, mint>;

#define _overloadvvm(_1, _2, _3, _4, name, ...) name

#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)



#define vvm1(a) vec<vm> a

#define vvm2(a, b) vec<vm> a(b)

#define vvm3(a, b, c) vec<vm> a(b,vm(c))

#define vvm4(a, b, c, d) vec<vm> a(b,vm(c,d))

#define vvm(...) _overloadvvm(__VA_ARGS__,vvm4,vvm3,vvm2 ,vvm1)(__VA_ARGS__)





//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); }

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



//便利関数



//テスト用

char ranc() {

    return (char) ('a' + rand() % 26);

}

int rand(int min, int 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);

        }

    }

}

vi ranv(int n, int min, int max) {

    vi v(n);

    rep(i, n)v[i] = rand(min, max);

    return v;

}

//単調増加

vi ranvi(int n, int min, int 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;

}

void ranvlr(int n, int min, int 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]);

    }

}

//便利 汎用

vp run_length(vi &a) {

    vp 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, int>> run_length(string &a) {

    vector<pair<char, 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;

}

template<class F> inline int mgr(int ok, int ng, F f) {

#define _mgrbody int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid;

    if (ok < ng)while (ng - ok > 1) { _mgrbody } else while (ok - ng > 1) { _mgrbody }

    return ok;

}



template<class F> inline int mgr(int ok, int ng, int second, F f) {

#define _mgrbody2 int mid = (ok + ng) / 2; if (f(mid, second))ok = mid; else ng = mid;

    if (ok < ng) while (ng - ok > 1) { _mgrbody2 } else while (ok - ng > 1) { _mgrbody2 }

    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 {

        rep(i, sz(a)) {

            if (a[i] < b[i]) {

                return b;

            } else if (a[i] > b[i])return a;

        }

    }

    return a;

}

//strを整数として比較

string smin(str &a, str b) {

    if (sz(a) < sz(b)) {

        return a;

    } else if (sz(a) > sz(b)) {

        return b;

    } else {

        rep(i, sz(a)) {

            if (a[i] < b[i]) {

                return a;

            } else if (a[i] > b[i])return b;

        }

    }

    return a;

}

template<typename V, typename T> int find(vector<V> &a,

        const T key

) {

    rep(i, sz(a))if (a[i] == key)

            return

                    i;

    return -1;

}

template<typename V, typename T> P find(vector<vector<V >> &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);

}

template<typename V, typename U> T find(vector<vector<vector<V >>> &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);

}



template<typename V, typename T> int count2(V &a, const T k) { return a == k; }

template<typename V, typename T> int count2(vector<V> &a, const T k) {

    int ret = 0;

    fora(v, a)ret +=

                      count(v, k

                      );

    return

            ret;

}

template<typename V, typename T> int count(vector<V> &a, const T k) {

    int ret = 0;

    fora(v, a)ret +=

                      count2(v, k

                      );

    return

            ret;

}

int count(str a, str k) {

    int ret = 0, len = k.length();

    auto pos = a.find(k);

    while (pos != string::npos)pos = a.find(k, pos + len), ret++;

    return ret;

}

template<typename V> int count_odd(V &a) { return a % 2; }

template<typename V> int count_odd(vector<V> &a) {

    int ret = 0;

    fora(v, a)ret += count_odd(v);

    return ret;

}

template<typename V> int count_even(V &a) { return a % 2 == 0; }

template<typename V> int count_even(vector<V> &a) {

    int ret = 0;

    fora(v, a)ret += count_even(v);

    return ret;

}

//algorythm

void iota(vector<int> &ve, int

s,

        int n

) {

    ve.

            resize(n);

    iota(all(ve), s

    );

}

vi iota(int s, int len) {

    vi ve(len);

    iota(all(ve), s);

    return ve;

}



inline ll rev(ll a) {

    ll res = 0;

    while (a) {

        res *= 10;

        res += a % 10;

        a /= 10;

    }

    return res;

}

template<class T> void rev(vector<T> &a) {

    reverse(all(a));

}

template<class T> vector<T> revv(vector<T> &a) {

    vector<T> b = a;

    reverse(all(b));

    return b;

}

void inline rev(string &a) {

    reverse(all(a));

}

constexpr int p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};



int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }



int keta(ll v) { if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) 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 if (v < p10[15])return 15; else return 16; } else { if (v < p10[17])return 17; else return 18; }}}}



int dsum(int v) {

    int ret = 0;

    for (; v; v /= 10)ret += v % 10;

    return ret;

}

inline int sumd(int v) { return dsum(v); }



struct sint {

    int v;

    sint(int v) : v(v) {}

    operator int() { return v; }

    int operator[](int i) { return (v / p10[i]) % 10; }

    int top(int i) {

        int len = keta(v);

        return operator[](len - 1 - i);

    }

    //先頭からi番目にセット

    int top(int i, int k) {

        int len = keta(v);

        return set(len - 1 - i, k);

    }

    int set(int i, int k) {

        if (i < 0)return top(abs(i) - 1, k);

        return v += p10[i] * (k - (v / p10[i]) % 10);

    }

    int inc(int i, int k = 1) {

        return v += p10[i] * k;

    }

    int inct(int i, int k = 1) {

        return v += p10[keta(v) - i - 1] * k;

    }

    int dec(int i, int k = 1) {

        return v -= p10[i] * k;

    }

    int dect(int i, int k = 1) {

        return v -= p10[keta(v) - i - 1] * k;

    }

#define op(t, o)template<class T>inline t operator o(T r){return v o r;}

    op(int, +=);

    op(int, -=);

    op(int, *=);

    op(int, /=);

    op(int, %=);

    op(int, +);

    op(int, -);

    op(int, *);

    op(int, /);

    op(int, %);

    op(bool, ==);

    op(bool, !=);

    op(bool, <);

    op(bool, <=);

    op(bool, >);

    op(bool, >=);

#undef op

    template<class T> inline int operator<<=(T r) { return v *= p10[r]; }

    template<class T> inline int operator<<(T r) { return v * p10[r]; }

    template<class T> inline int operator>>=(T r) { return v /= p10[r]; }

    template<class T> inline int operator>>(T r) { return v / p10[r]; }

};

int mask10(int v) { return p10[v] - 1; }

//変換系

template<class T, class U> inline vector<T> keys(vector<pair<T, U>> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}

template<class T, class U> inline vector<U> keys(map<T, U> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}



template<class T, class U> inline vector<U> keys(umap<T, U> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.fi);

    return res;

}



template<class T, class U> vector<U> inline values(vector<pair<T, U>> a) {

    vector<U> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline vector<T> values(map<T, U> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline vector<T> values(umap<T, U> a) {

    vector<T> res;

    for (auto &&k :a)res.pb(k.se);

    return res;

}



template<class T, class U> inline bool chma(T &a, const U &b) {

    if (a < b) {

        a = b;

        return true;

    }

    return false;

}

template<class U> inline bool chma(const U &b) { return chma(ma, b); }

template<class T, class U> inline bool chmi(T &a, const U &b) {

    if (b < a) {

        a = b;

        return true;

    }

    return false;

}

template<class U> inline bool chmi(const U &b) { return chmi(mi, b); }

template<class T> inline T min(T a, signed b) { return a < b ? a : b; }

template<class T> inline T max(T a, signed b) { return a < b ? b : a; }

template<class T> inline T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }

template<class T> inline T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }

template<class T> inline T min(vector<T> a) { return *min_element(all(a)); }

template<class T> inline T mini(vector<T> a) { return min_element(all(a)) - a.begin(); }

template<class T> inline T min(vector<T> a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T min(vector<T> a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a) { return *max_element(all(a)); }

template<class T> inline T maxi(vector<T> a) { return max_element(all(a)) - a.begin(); }

template<class T> inline T max(vector<T> a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a, int s, int n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<typename A, size_t N> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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> inline 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> inline A min(A (&a)[N][O]) {

    A res = min(a[0]);

    rep(i, N)res = min(res, max(a[i]));

    return res;

}

template<typename A, size_t N, size_t O, size_t P> inline 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> inline 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> inline 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> inline 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> inline T sum(T &v) { return v; }

template<class T> inline auto sum(vector<T> &v) {

    auto ret = sum(v[0]);

    rep(i, 1, sz(v))ret += sum(v[i]);

    return ret;

}

template<class T> inline 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> inline 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, class U, class... V> inline auto sum(vector<T> &v, U head, V... tail) {

    auto ret = sum(v[0], tail...);

    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);

    return ret;

}



inline P sump(vp &v, int len = -1) {

    if (len == -1)len = v.size();

    P res = {0, 0};

    chmi(len, v.size());

    rep(i, len) {

        res.fi += v[i].fi;

        res.se += v[i].se;

    }

    return res;

}



///要素が0の時、返り値は0か1か

template<class T> inline T mul(vector<T> &v, int

len = -1

) {

    if (len == -1)

        len = v.size();

    T res = 1;

    chmi(len, v

            .

                    size()

    );

    rep(i, len)res *= v[i];

    return

            res;

}

inline void clear(PQ &q) {q=PQ();}

template<class T> inline void clear(queue<T> &q) { while (q.size())q.pop(); }

template<class T> inline T *negarr(int size) {

    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));

    return body + size;

}

template<class T> inline T *negarr2(int h, int w) {

    double **dummy1 = new double *[2 * h + 1];

    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];

    dummy1[0] = dummy2 + w;

    for (int 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> inline 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> inline vi imo(vector<T> &a, U

k) {

    vector<T> ret = a;

    rep(i, sz(ret) - 1)ret[i + 1] = ret[i] + (a[i] == k);

    return

            ret;

}

template<class T> inline 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> inline 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> struct ruiC {

    const vector<T> rui;

    ruiC(vector<T> &ru) : rui(ru) {}

    inline T operator()(int l, int r) {

        assert(l <= r);

        return rui[r] - rui[l];

    }

    inline T operator[](int i) {

        return rui[i];

    }

};



template<class T> struct rruic {

    const T *rrui;

    rruic(T *ru) : rrui(ru) {}

    //n-1から-1へ

    inline T operator()(int l, int r) {

        assert(l >= r);

        return rrui[r] - rrui[l];

    }

    inline T operator[](int i) { return rrui[i]; }

};

template<class T> inline 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> inline ruiC<T> ruic(vector<T> &a) {

    vector<T> ret = ruiv(a);

    return ruiC<T>(ret);

}

//kと同じものの数

template<class T, class U> inline 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> inline ruiC<int> ruic(T &a, U k) {

    vi ret = ruiv(a, k);

    return ruiC<int>(ret);

}



//xor

template<class T> inline 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> inline 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> inline vector<T> ruimi(vector<T> &a) {

    int 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> rruic<T> rrui(vector<T> &a) {

    int len = a.size();

    T *body = (T *) malloc((len + 1) * sizeof(T));

    T *res = body + 1;

    rer(i, len - 1)res[i - 1] = res[i] + a[i];

    return rruic<T>(res);

}

//掛け算

template<class T> T *rruim(vector<T> &a) {

    int 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(T &a, U v = 1) { a += v; }

template<class T, class U> void inc(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        inc(u, v

        );

}

template<class T> void inc(vector<T> &a) { for (auto &u :a)inc(u, 1); }

template<class T, class U> void plus(T &a, U v = 1) { a += v; }

template<class T, class U> void plus(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        inc(u, v

        );

}

template<class T> void plus(vector<T> &a) { for (auto &u :a)inc(u, 1); }

template<class T, class U> void dec(T &a, U v = 1) { a -= v; }

template<class T, class U> void dec(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        dec(u, v

        );

}

template<class T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }

template<class T, class U> void minu(T &a, U v = 1) { a -= v; }

template<class T, class U> void minu(vector<T> &a, U

v = 1

) {

    for (

        auto &u

            :a)

        dec(u, v

        );

}

template<class T> void minu(vector<T> &a) { for (auto &u :a)dec(u, 1); }

template<class U> void minu(string &a, U v = 1) { for (auto &u :a)dec(u, v); }

void minu(string &a) { for (auto &u :a)dec(u, 1); }

inline bool inside(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }

inline bool inside(int l, int v, int r) { return l <= v && v < r; }

#define ins inside

ll u(ll a) { return a < 0 ? 0 : a; }

template<class T> vector<T> u(const vector<T> &a) {

    vector<T> ret = a;

    fora(v, ret)v = u(v);

    return ret;

}

#define MIN(a) numeric_limits<a>::min()

#define MAX(a) numeric_limits<a>::max()

ll goldd(ll left, ll right, function<ll(ll)> 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;

}

ll goldt(ll left, ll right, function<ll(ll)> 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;

}

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;

}



constexpr bool bget(ll m, int keta) {

    assert(keta < 64);

    return (m >> keta) & 1;

}

int bget(ll m, int keta, int sinsuu) {

    m /= (ll) pow(sinsuu, keta);

    return m % sinsuu;

}

ll bit(int n) { return (1LL << (n)); }

ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }

int mask(int n) { return (1ll << n) - 1; }

//int bcou(ll m) { return __builtin_popcount(m & 0xFFFFFFFF) + __builtin_popcount(m >> 32); }

#define bcou __builtin_popcountll

//最下位ビット

int lbit(int n) {

    return n & -n;

}

//最上位ビット

int hbit(int n) {

    n |= (n >> 1);

    n |= (n >> 2);

    n |= (n >> 4);

    n |= (n >> 8);

    n |= (n >> 16);

    n |= (n >> 32);

    return n - (n >> 1);

}

int hbitk(int n) {

    int k = 0;

    rer(i, 5) {

        int a = k + (1ll << i);

        int b = 1ll << a;

        if (b <= n)k += 1ll << i;

    }

    return k;

}

//初期化は0を渡す

ll nextComb(ll &mask, int n, int 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

vl bitCombList(int n, int r) {

    vl res;

    int m = 0;

    while (nextComb(m, n, r)) {

        res.pb(m);

    }

    return res;

}

//大文字小文字を区別する

int altoiaZ(char c) {

    if ('A' <= c && c <= 'Z')return c - 'A';

    return c - 'a' + 26;

}

char itoalaZ(int i) {

    if (i < 26)return 'A' + i;

    return 'a' + i - 26;

}

//aもAも0を返す 基本小文字

int altoi(char c) {

    if ('A' <= c && c <= 'Z')return c - 'A';

    return c - 'a';

}

char itoal(int i) {

    return 'a' + i;

}

int ctoi(char c) { return c - '0'; }

char itoc(int i) { return i + '0'; }

int vtoi(vi &v) {

    int 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(int i) {

    vi res;

    while (i) {

        res.pb(i % 10);

        i /= 10;

    }

    rev(res);

    return res;

}

vi stov(string &a) {

    int n = sz(a);

    vi ret(n);

    rep(i, n) {

        ret[i] = a[i] - '0';

    }

    return ret;

}

vector<vector<int>> ctoi(vector<vector<char>> s, char c) {

    int n = sz(s), m = sz(s[0]);

    vector<vector<int>> res(n, vector<int>(m));

    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;

    return res;

}

#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );



//[i] := i番として圧縮されたものを返す

vi compress(vi &a) {

    vi b;

    int len = a.size();

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) {

        a[i] = lower_bound(all(b), a[i]) - b.begin();

    }

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

vi compress(vi &a, umap<int, int> &map) {

    vi b;

    int len = a.size();

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) {

        int v = a[i];

        a[i] = lower_bound(all(b), a[i]) - b.begin();

        map[v] = a[i];

    }

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}



vi compress(vi &a, vi &r) {

    vi b;

    int len = a.size();

    fora(v, a)b.pb(v);

    fora(v, r)b.pb(v);

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();

    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();

    int 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;

    int len = a.size();

    fora(v, a)b.pb(v);

    fora(v, r)b.pb(v);

    fora(v, s)b.pb(v);

    sort(b);

    unique(b);

    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();

    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();

    for (int i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

vi compress(_vvi &a) {

    vi b;

    fora(vv, a)fora(v, vv)b.pb(v);

    sort(b);

    unique(b);

    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();

    int 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.pb(v);

    sort(b);

    unique(b);

    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();

    int blen = sz(b);

    vi ret(blen);

    rep(i, blen) {

        ret[i] = b[i];

    }

    return ret;

}

void compress(int a[], int len) {

    vi b;

    for (int i = 0; i < len; ++i) {

        b.push_back(a[i]);

    }

    sort(b);

    unique(b);

    for (int 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 lowerBound(a, v) (*lower_bound(all(a),v))

#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())

#define upperBound(a, v) (*upper_bound(all(a),v))

#define ans(a) cout<<a<<endl;continue;

#define poll(a) q.front();q.pop()

#define dpoll(a) q.front();q.pop_front()

#define pollLast(a) q.back();q.pop_back()

#define pollBack(a) q.back();q.pop_back()

template<class T> inline void fin(T s) { cout << s << endl, exit(0); }



//便利 数学 math

int mod(int a, int m) { return (a % m + m) % m; }

int pow(int a) { return a * a; };

ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }

ll comi(int n, int 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 comd(int n, int r) {

    assert(n < 2020);

    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];

}

int gcd(int a, int b) {

    while (b) a %= b, swap(a, b);

    return abs(a);

}

int 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) {

    int 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;

    }

}

//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); }

ll sig(ll t) { return (1 + t) * t / 2; }

ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }





namespace pnumber_tools {

#define Eratosthenes30_2  class Eratosthenes {public:virtual void initialize() {} virtual void generate(const int64_t x) = 0;        virtual void generate(const int64_t from, const int64_t to) = 0;        virtual int64_t count() = 0;        virtual bool prime(int x) = 0;        virtual int32_t version() = 0;        inline uint64_t PopCnt(uint64_t);        inline int BitToIndex(uint8_t);    };    uint64_t Eratosthenes::PopCnt(uint64_t bits) {        bits = (bits & 0x5555555555555555ULL) + ((bits >> 1) & 0x5555555555555555ULL);        bits = (bits & 0x3333333333333333ULL) + ((bits >> 2) & 0x3333333333333333ULL);        bits = (bits & 0x0f0f0f0f0f0f0f0fULL) + ((bits >> 4) & 0x0f0f0f0f0f0f0f0fULL);        return (bits * 0x0101010101010101ULL) >> (64 - 8);    }    int Eratosthenes::BitToIndex(uint8_t b) {        switch (b) {            case 1 << 0:                return 0;            case 1 << 1:                return 1;            case 1 << 2:                return 2;            case 1 << 3:                return 3;            case 1 << 4:                return 4;            case 1 << 5:                return 5;            case 1 << 6:                return 6;            case 1 << 7:                return 7;        }        return -1;    }    namespace {        const uint64_t kSegmentSize = 1000000;          const int32_t kMod30[] = {1, 7, 11, 13, 17, 19, 23, 29};        const int C1[] = {6, 4, 2, 4, 2, 4, 6, 2};        const int C0[][8] = {                {0, 0, 0, 0, 0, 0, 0, 1},                {1, 1, 1, 0, 1, 1, 1, 1},                {2, 2, 0, 2, 0, 2, 2, 1},                {3, 1, 1, 2, 1, 1, 3, 1},                {3, 3, 1, 2, 1, 3, 3, 1},                {4, 2, 2, 2, 2, 2, 4, 1},                {5, 3, 1, 4, 1, 3, 5, 1},                {6, 4, 2, 4, 2, 4, 6, 1},        };        const uint8_t kMask[][8] = {                {0xfe, 0xfd, 0xfb, 0xf7, 0xef, 0xdf, 0xbf, 0x7f},                {0xfd, 0xdf, 0xef, 0xfe, 0x7f, 0xf7, 0xfb, 0xbf},                {0xfb, 0xef, 0xfe, 0xbf, 0xfd, 0x7f, 0xf7, 0xdf},                {0xf7, 0xfe, 0xbf, 0xdf, 0xfb, 0xfd, 0x7f, 0xef},                {0xef, 0x7f, 0xfd, 0xfb, 0xdf, 0xbf, 0xfe, 0xf7},                {0xdf, 0xf7, 0x7f, 0xfd, 0xbf, 0xfe, 0xef, 0xfb},                {0xbf, 0xfb, 0xf7, 0x7f, 0xfe, 0xef, 0xdf, 0xfd},                {0x7f, 0xbf, 0xdf, 0xef, 0xf7, 0xfb, 0xfd, 0xfe},        };        const int64_t kSievedPrimes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23};        const int64_t kInitialSize = 7 * 11 * 13 * 17 * 19 * 23;class Eratosthenes2 final : public Eratosthenes {    public:        void generate(const int64_t x) override;        void generate(const int64_t, const int64_t) override { assert(false); }        bool prime(int x) override;        int64_t count() override;        int32_t version() override { return 2; }        std::vector<uint8_t> flags_;    };    void Eratosthenes2::generate(const int64_t x) {        flags_.clear();        if (x > 10000000000)            return;        int64_t size = x / 30 + (x % 30 != 0);        flags_.resize(size, 0xff);        if (int r = x % 30) {            if (r <= 1) flags_[size - 1] = 0x0;            else if (r <= 7) flags_[size - 1] = 0x1;            else if (r <= 11) flags_[size - 1] = 0x3;            else if (r <= 13) flags_[size - 1] = 0x7;            else if (r <= 17) flags_[size - 1] = 0xf;            else if (r <= 19) flags_[size - 1] = 0x1f;            else if (r <= 23) flags_[size - 1] = 0x3f;            else if (r <= 29) flags_[size - 1] = 0x7f;        }        flags_[0] = 0xfe;        const uint64_t sqrt_x = std::ceil(std::sqrt(x) + 0.1);        const uint64_t sqrt_xi = sqrt_x / 30 + 1;        for (uint64_t i = 0; i < sqrt_xi; ++i) {            for (uint8_t flags = flags_[i]; flags; flags &= flags - 1) {                uint8_t lsb = flags & (-flags);                int ibit = BitToIndex(lsb);                const int32_t m = kMod30[ibit];                int32_t pm = 30 * i + 2 * m;                for (uint64_t j = i * pm + (m * m) / 30, k = ibit; j < flags_.size();                        j += i * C1[k] + C0[ibit][k], k = (k + 1) & 7) {                    flags_[j] &= kMask[ibit][k];                }            }        }    }int64_t Eratosthenes2::count() {            if (flags_.empty())                return -1;            int64_t ret = 3;            for (uint8_t f : flags_) {                ret += PopCnt(f);            }            return ret;        }        inline bool Eratosthenes2::prime(int x) {            static unordered_map<int, int> imap = {{1,  0},                                                   {7,  1},                                                   {11, 2},                                                   {13, 3},                                                   {17, 4},                                                   {19, 5},                                                   {23, 6},                                                   {29, 7}};            if (x == 2 || x == 3 || x == 5)return true;            else if (!(x & 1) || x % 3 == 0 || x % 5 == 0)return false;            return (flags_[x / 30] >> imap[x % 30]) & 1;        }}

    Eratosthenes30_2

    Eratosthenes2 era;

    int prime_len = 3e6;

    vector<int> primes;

    void setprime(int size = 3e6) {

        prime_len = size;

        era.generate(size);

        primes = {2, 3, 5};

        int nums[129] = {};

        nums[1] = 1;

        nums[2] = 7;

        nums[4] = 11;

        nums[8] = 13;

        nums[16] = 17;

        nums[32] = 19;

        nums[64] = 23;

        nums[128] = 29;

        //p = 30i + j

        int j = 0;

        int sflim = era.flags_.size();

        int i30 = 0;

        for (int i = 0; i < sflim; i++, i30 += 30) {

            uint8_t m = era.flags_[i];

            while (m) {

                j = m & -m;

                primes.push_back(i30 + nums[j]);

                m ^= j;

            }

        }

    }

    using u32 = unsigned int;//longlongになっていると思うが放置していいと思う

    using u64 = unsigned long long;

    using u128 = __uint128_t;



    //O(k log(n)^3)

    //32bit -> 270回程度

    //64bit -> 1260回程度

    //64bit以下で精度100%

    template<class BinOp> bool miller_rabin(const uint64_t &n, const uint64_t *witness, BinOp modmul) {

        if (n == 2) return true;

        if (n < 2 || n % 2 == 0) return false;

        const int64_t m = n - 1, d = m / (m & -m);

        auto modpow = [&](int64_t a, int64_t b) {

            int64_t res = 1;

            for (; b; b /= 2) {

                if (b & 1) res = modmul(res, a);

                a = modmul(a, a);

            }

            return res;

        };

        auto suspect = [&](uint64_t a, uint64_t t) {

            a = modpow(a, t);

            while (t != n - 1 && a != 1 && a != n - 1) {

                a = modmul(a, a);

                t = modmul(t, 2);

            }

            return a == n - 1 || t % 2 == 1;

        };

        for (const uint64_t *w = witness; *w; w++) {

            if (*w % n != 0 && !suspect(*w, d)) return false;

        }

        return true;

    }



//miller_rabin & エラトステネス

    bool prime(const int &n) {

        assert(n < 9223372036854775808ULL);

        if (!sz(primes))setprime();

        //エラトステネスの篩 偶数省略ver

        //[i] = i*2+1

        if (n < prime_len)return era.prime(n);

        if (n < 4294967296) {

            rep(i, 30) if (!(n % primes[i]))return false;

            constexpr uint64_t witness[] = {2, 7, 61, 0};

            auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return a * b % n; };

            return miller_rabin(n, witness, modmul);

        } else {

            rep(i, 50) if (!(n % primes[i]))return false;

            constexpr uint64_t witness[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022, 0};

            auto modmul = [&](uint64_t a, uint64_t b) -> uint64_t { return (uint64_t) ((__uint128_t) a * b % n); };

            return miller_rabin(n, witness, modmul);

        }

    }

    vi divisors(int v) {

        vi res;

        double lim = std::sqrt(v);

        for (int i = 1; i <= lim; ++i) {

            if (v % i == 0) {

                res.pb(i);

//                if (i != v / i)res.pb(v / i);

            }

        }

        rer(i, sz(res) - 1) {

            if (res[i] != v / res[i])res.pb(v / res[i]);

        }

        return res;

    }

    vp factorization(int v) {

        assert(v);

        int tv = v;

        vp res;

        if (!sz(primes))setprime();

        for (auto &&p :primes) {

            if (v % p == 0)res.eb(p, 0);

            while (v % p == 0) {

                v /= p;

                res.back().second++;

            }

            if (v == 1 || p * p > tv)break;

        }

        if (v > 1)res.eb(v, 1);

        return res;

    }



    //注意 gcdを取る間隔 i%10 が場合によってはtleを引き起こす

    //うまくいかなかったら%1で動かすように変更した

    vp rho(int v) {

        assert(v);

        if (v == 1)return vp();

        vi ret;

        if (!sz(primes)) setprime();

        int lim = 300;                      //64bit

        if (v < 4294967296)lim = 80;      //32bit

        else if (v < 281474976710656LL)lim = 120; //48bit

        for (int i = 0; i < lim; i += 5) {

            //v<=1の判定回数を減らせる分高速化できる事を期待している

            while (v % primes[i] == 0) ret.pb(primes[i]), v /= primes[i];

            while (v % primes[i + 1] == 0) ret.pb(primes[i + 1]), v /= primes[i + 1];

            while (v % primes[i + 2] == 0) ret.pb(primes[i + 2]), v /= primes[i + 2];

            while (v % primes[i + 3] == 0) ret.pb(primes[i + 3]), v /= primes[i + 3];

            while (v % primes[i + 4] == 0) ret.pb(primes[i + 4]), v /= primes[i + 4];

            if (v <= 1) {

                sort(ret);//試し割りだけで終わった時

                return run_length(ret);

            }

        }

        int lower_prime_bound = primes[lim - 1] * primes[lim - 1];



        //試し割りするか

        mt19937 mr;

        mr.seed(clock());

        int c;

        auto fac = [&](int v, const int loop = 10) {

//        if (v == 4)return 2ll; //試し割りで消えた

            c = mr() % v;

            int x = mr() % v;

            int y = x;

            int d = 1;

            int lim = std::sqrt(v) + 2;

            //brentによる改良 差分を一操作で求められる

            for (int i = 1, mul = 1, next_2pow = 2; d == 1 && i <= lim; i++) {

                if (i == next_2pow) {

                    y = x;

                    next_2pow <<= 1;

                }

                x = ((u128) x * x + c) % v;

                mul = ((u128) mul * abs(x - y)) % v;//ユークリッドの互除法と同じ原理で因数は保たれる

                if (i % loop == 0) {//10とかなんとか iによっては死ぬ

                    d = gcd(mul, v);

                    mul = 1;

                }

            }

            if (d == v) return -1ll;

            return d;

        };



        queue<int> q;

        q.push(v);

        while (!q.empty()) {

            int v = q.front();

            q.pop();

            if (prime(v)) {

                ret.pb(v);

                continue;

            }

            int d = -1;

            //基本loop回数10でやり、駄目なら1で

            for (int i = 0; d == -1 && i < 5; i++)d = fac(v, 10);

            if (d == -1)while (d == -1)d = fac(v, 1);

            int d2 = v / d;

            rep(_, 2) {

                if (d <= lower_prime_bound)ret.pb(d);

                else q.push(d);

                d = d2;

            }

        }

        sort(ret);

        return run_length(ret);

    }

}

using namespace pnumber_tools;

//幾何 Pをcomplexとして扱う

template<class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }

dou atan2(pd a) { return atan2(a.se, a.fi); }

dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }

dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }

//bを中心とするabcのtheta aからcにかけて時計回り

dou angle(pd a, pd b, pd c) {

    dou ax = a.fi - b.fi;

    dou ay = a.se - b.se;

    dou cx = c.fi - b.fi;

    dou cy = c.se - b.se;

    double ret = atan2(cy, cx) - atan2(ay, ax);

    if (ret < 0) ret += 2 * PI;

    return ret;

}



dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }

dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }



template<class T> struct edge {

    int f, t;

    T c;

    int id;

    int type;

    edge(int f, int t, T c = 1, int id = -1, int ty = -1) : f(f), t(t), c(c), id(id), type(ty) {}

    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 {

protected:

    vector<bool> _used;

public :

    vector<vector<edge<T>>> g;

    vector<edge<T>> edges;

    int n;

    graph(int n) : n(n) { g.resize(n), _used.resize(n); }

    void clear() { g.clear(), edges.clear(); }

    void resize(int n) {

        this->n = n;

        g.resize(n);

        _used.resize(n);

    }

    int size() { return g.size(); }

    vector<edge<T> > &operator[](int i) { return g[i]; }

    virtual void add(int f, int t, T c, int id, int ty) = 0;

    virtual bool used(edge<T> &e) = 0;

    virtual bool used(int id) = 0;

    virtual void del(edge<T> &e) = 0;

    virtual void del(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;

    using graph<T>::_used;

    int eid = 0;



    digraph(int n) : graph<T>(n) {}

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("digraph add");

            deb(f, t, c, id, ty);

            ole();

        }

        if (id == -1)id = eid++;

        g[f].emplace_back(f, t, c, id, ty);

        edges.emplace_back(f, t, c, id, ty);

    }

    bool used(edge<T> &e) { return _used[e.id]; }

    bool used(int id) { return _used[id]; }

    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }

    void del(int id) { _used[id] = _used[id ^ 1] = 1; }

    void set_edges() {

        if (sz(edges))return;

        rep(i, n)fora(e, g[i])edges.pb(e);

    }

};

template<class T=int> class undigraph : public graph<T> {

public:

    using graph<T>::g;

    using graph<T>::n;

    using graph<T>::edges;

    using graph<T>::_used;

    int eid = 0;

    undigraph(int n) : graph<T>(n) {

    }

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("undigraph add");

            deb(f, t, c, id, ty);

            ole();

        }

        if (id == -1)id = eid, eid += 2;

        g[f].emplace_back(f, t, c, id, ty);

        g[t].emplace_back(t, f, c, id + 1, ty);

        edges.emplace_back(f, t, c, id, ty);

        edges.emplace_back(t, f, c, id + 1, ty);

    }

    void add(edge<T> &e) {

        int f = e.f, t = e.t, ty = e.type;

        T c = e.c;

        add(f, t, c, ty);

    }

    bool used(edge<T> &e) { return _used[e.id]; }

    bool used(int id) { return _used[id]; }

    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }

    void del(int id) { _used[id] = _used[id ^ 1] = 1; }

    void set_edges() {

        if (sz(edges))return;

        rep(i, n)fora(e, g[i])edges.pb(e);

    }

};

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; }

    inline int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }

    void push(u64 key, const T &value) {

        ++size;

        v[getbit(key ^ last)].emplace_back(key, value);

    }

    pair<u64, T> pop() {

        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;

    }

};

vi dijkstra2(const graph<int> &g, int s, int cant_arrive = -1) {

    if (!(0 <= s && s < g.n)) {

        debugline("dijkstra");

        deb(s, g.n);

        ole();

    }

    int initValue = MAX(int);

    vi dis(g.n, initValue);

    radixheap<int> q;

    dis[s] = 0;

    q.push(0, s);

    while (!q.empty()) {

        int nowc, i;

        tie(nowc, i) = q.pop();

        if (dis[i] != nowc)continue;

        for (auto &&e  : g.g[i]) {

            int to = e.t;

            int c = nowc + e.c;

            if (dis[to] > c) {

                dis[to] = c;

                q.push(dis[to], to);

            }

        }

    }

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;

    return dis;

}



template<class T> vector<T> dijkstra(const graph<T> &g, int s, int cant_arrive = -1) {

    if (!(0 <= s && s < g.n)) {

        debugline("dijkstra");

        deb(s, g.n);

        ole();

    }

    T initValue = MAX(T);

    vector<T> dis(g.n, initValue);

    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;

                q.emplace(dis[to], to);

            }

        }

    }

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;

    return dis;

}

template<class T> vector<vector<T>> warshall(const graph<T> &g, int cant_arrive = -1) {

    int n = g.n;

    vector<vector<T> > dis(n, vector<T>(n, linf));

    rep(i, n)fora(e, g.g[i])chmi(dis[e.f][e.t], e.c);

    rep(i, n)dis[i][i] = 0;

    rep(k, n)rep(i, n)rep(j, n)chmi(dis[i][j], dis[i][k] + dis[k][j]);

    //基本、たどり着かないなら-1

    if (cant_arrive == -1)rep(i, n)rep(j, n) if (dis[i][j] == linf)dis[i][j] = -1;

    return dis;

}



template<class T=int> class tree : public undigraph<T> {

public:

    using undigraph<T>::g;

    using undigraph<T>::n;

    using undigraph<T>::edges;

    using undigraph<T>::_used;

    vi disv;

private:

    bool never = 1;

    int root = -1;

    vi dep;//到着した順の深さ

    vi vis;//到着した順

    vi first;//初めてiに到着した時

    vector<int> bestt;//タイミングを持つ

    int sn = 0;//seg木の底辺の長さ euler tourの長さ

    void built() {

        never = 0;

        n = g.size();

        //euler tour

        disv.resize(n);

        first.resize(n);

        dep.resize(n * 2);

        vis.resize(n * 2);

        function<void(int, int, int, int &)> dfs = [&](int i, int p, int d, int &k) {

            first[i] = k;

            vis[k] = i;

            dep[k++] = d;

            forg(gi, g[i]) {

                if (t == p)continue;

                disv[t] = disv[i] + c;

                dfs(t, i, d + 1, k);

                vis[k] = i;

                dep[k++] = d;

            }

        };

        int k = 0;

        dfs(root, -1, 0, k);

        //rmq

        sn = 1;

        while (sn < n * 2)sn <<= 1;



        bestt.resize((sn << 1) - 1);

        rep(i, k)bestt[i + sn - 1] = i;//時間を持つ

        rer(i, sn - 2) {

            int lt = (i << 1) + 1, rt = (i << 1) + 2;

            if (dep[bestt[lt]] < dep[bestt[rt]]) {

                bestt[i] = bestt[lt];

            } else {

                bestt[i] = bestt[rt];

            }

        }

    }

    //最小の深さを持つタイミングが知りたい

    inline int lca_rmq(int l, int r) {

        int midv = inf;

        int midi = -1;

        l += sn - 1;

        r += sn - 1;

        while (l < r) {

            //右なら、次に右上へ移動してしまうので

            if (!(l & 1)) {

                if (chmi(midv, dep[bestt[l]])) {

                    midi = bestt[l];

                }

            }

            //右なら、左を取らないと機会を逃す

            if (!(r & 1)) {

                if (chmi(midv, dep[bestt[r - 1]])) {

                    midi = bestt[r - 1];

                }

            }

            l >>= 1;

            r = (r - 1) >> 1;

        }

        return vis[midi];

    }



public:

    tree(int n, int root = 0) : undigraph<T>(n), root(root) {}

    bool leaf(int v) {

        return sz(g[v]) == 1 && v != root;

    }

    int lca(int u, int v) {

        if (never)built();

        if (first[u] > first[v])swap(u, v);

        return lca_rmq(first[u], first[v] + 1);

    }

    int dis(int u, int v) {

        if (never) built();

        if (first[u] > first[v])swap(u, v);

        int p = lca_rmq(first[u], first[v] + 1);

        return disv[u] + disv[v] - disv[p] * 2;

    }



};

//辺によりメモリを大量消費

// よってedgesを消している

//頂点10^6でメモリを190MB(制限の8割)使う



//軽量化のため、辺を持たないbig gridクラスがあってもいいかもしれない

//

template<class T=int> class grid_k6 : public undigraph<T> {

public:

    using undigraph<T>::g;

    using undigraph<T>::n;

    using undigraph<T>::edges;

    using undigraph<T>::_used;

    int H, W;

    int eid = 0;

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("grid_k6 add");

            deb(f, t, c, id, ty);

            ole();

        }

        g[f].emplace_back(f, t, c, eid++, ty);

        g[t].emplace_back(t, f, c, eid++, ty);

    }

    int getid(int h, int w) {

        if (!inside(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);

    }

    grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {

        rep(h, H) {

            rep(w, W) {

                int f = getid(h, w);

                if (w + 1 < W) add(f, getid(h, w + 1));

                if (h + 1 < H)add(f, getid(h + 1, w));

            }

        }

    }

    grid_k6(_vvc 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));

                }

            }

        }

    }

    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.pb(e);

    }

};

//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;

    using digraph<T>::_used;

    int H, W;

    int eid = 0;

    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {

        if (!(0 <= f && f < n && 0 <= t && t < n)) {

            debugline("digrid_k6 add");

            deb(f, t, c, id, ty);

            ole();

        }

        g[f].emplace_back(f, t, c, eid++, ty);

    }

    int getid(int h, int w) {

        if (!inside(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) {

        rep(h, H) {

            rep(w, W) {

                int f = getid(h, w);

                if (w + 1 < W) add(f, getid(h, w + 1));

                if (h + 1 < H)add(f, getid(h + 1, w));

            }

        }

    }

    digrid_k6(_vvc 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.pb(e);

    }

};

template<class T> bool nibu(const 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 1;

}

//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;



}

template<class T> ostream &operator<<(ostream &os, graph<T> &g) {

    rep(i, sz(g)) {

        forg(gi, g[i]) {

            os << f << " " << t << " " << c << endl;

        }

    }

    return os;

}



//機能拡張

vp vtop(vi &a, vi &b) {

    vp res(sz(a));

    rep(i, sz(a))res[i] = mp(a[i], b[i]);

    return res;

}

void ptov(vp &p, vi &a, vi &b) {

    a.resize(sz(p));

    b.resize(sz(p));

    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;

}

template<typename _CharT, typename _Traits, typename _Alloc>

basic_string<_CharT, _Traits, _Alloc>

operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {

    basic_string<_CharT, _Traits, _Alloc> __str(__lhs);

    __str.append(to_string(__rv));

    return __str;

}

template<typename _CharT, typename _Traits, typename _Alloc>

void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const int __rv) {

    __lhs += to_string(__rv);

}

template<typename _CharT, typename _Traits, typename _Alloc>

basic_string<_CharT, _Traits, _Alloc>

operator+(const basic_string<_CharT, _Traits, _Alloc> &__lhs, const signed __rv) {

    basic_string<_CharT, _Traits, _Alloc> __str(__lhs);

    __str.append(to_string(__rv));

    return __str;

}

template<typename _CharT, typename _Traits, typename _Alloc>

void operator+=(basic_string<_CharT, _Traits, _Alloc> &__lhs, const signed __rv) {

    __lhs += to_string(__rv);

}

template<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, class U>

priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, vector <U> &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, class U> set<T> &operator+=(set<T> &a, U v) {

    a.insert(v);

    return a;

}

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.

            pb(v);

    return

            a;

}

template<class T, class U> vector<T> operator+(const vector <T> &a, U v) {

    vector<T> ret = a;

    ret += v;

    return ret;

}

template<class T, class U> vector<T> operator+(U v, const vector <T> &a) {

    vector<T> ret = a;

    ret.insert(ret.begin(), v);

    return ret;

}

template<class T> vector<T> operator+(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) {

    fora(v, b)a += v;

    return a;

}

template<class T, class U> vector<T> &operator+=(vector<T> &a, initializer_list<U>

v) {

    for (

        auto &&va

            :v)

        a.

                pb(va);

    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();

    }

    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<typename T> void remove(vector<T> &v, unsigned int

i) {

    v.

            erase(v

                          .

                                  begin()

                  + i);

}

template<typename T> void remove(vector<T> &v, unsigned int

s,

        unsigned int e

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + e);

}

template<typename T> void removen(vector<T> &v, unsigned int

s,

        unsigned int n

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + s + n);

}

template<typename T> void erase(vector<T> &v, unsigned int

i) {

    v.

            erase(v

                          .

                                  begin()

                  + i);

}

template<typename T> void erase(vector<T> &v, unsigned int

s,

        unsigned int e

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + e);

}

template<class T, class U> void erase(map<T, U> &m, int okl, int ngr) {

    m.erase(m.lower_bound(okl), m.lower_bound(ngr));

}

template<class T> void erase(set<T> &m, int okl, int ngr) {

    m.erase(m.lower_bound(okl), m.lower_bound(ngr));

}

template<typename T> void erasen(vector<T> &v, unsigned int

s,

        unsigned int n

) {

    v.

            erase(v

                          .

                                  begin()

                  + s, v.

            begin()

                       + s + n);

}

template<typename T, typename U> void insert(vector<T> &v, unsigned int

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 int

i,

        vector<T> list

) {

    for (

        auto &&va

            :list)

        v.

                insert(v

                               .

                                       begin()

                       + i++, va);

}

template<typename T, typename U> void insert(vector<T> &v, initializer_list<U>

list) {

    for (

        auto &&va

            :list)

        v.

                pb(va);

}

template<typename T, typename U> void insert(vector<T> &v, unsigned int

i,

        initializer_list<U> 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<typename T> void insert(set<T> &v, initializer_list<T> list) {

    for (auto &&va :list)v.insert(va);

}





//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.pb(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.pb(i);

        fora(e, g[i])if (--nyu[e.t] == 0)q.push(e.t);

    }

    return res.size() == n;

}

vector<string> split(const string a, const char deli) {

    string b = a + deli;

    int 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) {

    string b = a + deli;

    int l = 0, r = 0, n = b.size(), dn = deli.size();

    vector<string> res;

    rep(i, n) {

        if (i + dn <= n && b.substr(i, i + dn) == deli) {

            r = i;

            if (l < r)res.push_back(b.substr(l, r - l));

            i += dn - 1;

            l = i + 1;

        }

    }

    return res;

}

void yn(bool a) {

    if (a)cout << "yes" << endl;

    else cout << "no" << endl;

}

void Yn(bool a) {

    if (a)cout << "Yes" << endl;

    else cout << "No" << endl;

}

void YN(bool a) {

    if (a)cout << "YES" << endl;

    else cout << "NO" << endl;

}

void fyn(bool a) {

    if (a)cout << "yes" << endl;

    else cout << "no" << endl;

    exit(0);

}

void fYn(bool a) {

    if (a)cout << "Yes" << endl;

    else cout << "No" << endl;

    exit(0);

}

void fYN(bool a) {

    if (a)cout << "YES" << endl;

    else cout << "NO" << endl;

    exit(0);

}

void Possible(bool a) {

    if (a)cout << "Possible" << endl;

    else cout << "Impossible" << endl;

    exit(0);

}

void POSSIBLE(bool a) {

    if (a)cout << "POSSIBLE" << endl;

    else cout << "IMPOSSIBLE" << endl;

    exit(0);

}



int n, m, k, d, H, W, x, y, z, q;

int cou;

vi t, a, b, c;

//vvi (s, 0, 0);

vvc (ba, 0, 0);

vp p;

str s;



void solve() {

    cin >> n >> k;

    na(a, n);

    rep(300)

    {

        priority_queue<int, vector<int>, greater<int> > q;//小さい順

        rep(l, k + 1) {

            rep(r, k - l + 1) {

                if (l + r > n)break;

                int del = k - l - r;

//                q = PQ();

                while(sz(q))q.pop();

                int su = 0;

                rep(i, l) {

                    su += a[i];

                    q += a[i];

                }

                rep(i, r) {

                    su += a[n - 1 - i];

                    q += a[n - 1 - i];

                }

                rep(i, del) {

                    if (q.empty())break;

                    int d = q.top();

                    if (d >= 0)break;

                    q.pop();

                    su -= d;

                }

                chma(su);

            }

        }

    }

    cout << ma << endl;



}



int my(int n, vi &a) {

    return 0;

}

int sister(int n, vi &a) {

    int ret = 0;



    return ret;

}



signed main() {

    solve();



#define arg n,a

#ifdef _DEBUG

    bool bad = 0;

    for (int i = 0, ok = 1; i < k5 && ok; i++) {

        int n = rand(1, 8);

        vi a = ranv(n, 1, 10);

        int myres = my(arg);

        int res = sister(arg);

        ok = myres == res;

        if (!ok) {

            out(arg);

            cerr << "正解 : " << res << endl;

            cerr << "自分  : " << 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;

    }

#endif

    return 0;

};