Problem:
Score : 700 points

Problem Statement

We will define the median of a sequence b of length M, as follows:

Let b' be the sequence obtained by sorting b in non-decreasing order. Then, the value of the (M / 2 + 1)-th element of b' is the median of b. Here, / is integer division, rounding down.

For example, the median of (10, 30, 20) is 20; the median of (10, 30, 20, 40) is 30; the median of (10, 10, 10, 20, 30) is 10.

Snuke comes up with the following problem.

You are given a sequence a of length N.
For each pair (l, r) (1 \leq l \leq r \leq N), let m_{l, r} be the median of the contiguous subsequence (a_l, a_{l + 1}, ..., a_r) of a.
We will list m_{l, r} for all pairs (l, r) to create a new sequence m.
Find the median of m.

Constraints

1 \leq N \leq 10^5

a_i is an integer.

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

Print the median of m.

Sample Input 1

3
10 30 20

Sample Output 1

30

The median of each contiguous subsequence of a is as follows:

The median of (10) is 10.

The median of (30) is 30.

The median of (20) is 20.

The median of (10, 30) is 30.

The median of (30, 20) is 30.

The median of (10, 30, 20) is 20.

Thus, m = (10, 30, 20, 30, 30, 20) and the median of m is 30.

Sample Input 2

1
10

Sample Output 2

10

Sample Input 3

10
5 9 5 9 8 9 3 5 4 3

Sample Output 3

8

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

//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 _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,)(__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 fora(a, b) for(auto&& a : b)
#define forg(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++)
#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++)

//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 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<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); }
template<class T> void replace(str &a, T key, T v) { a.replace(all(a), key, v); }
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;
    }
};
//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 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__)

#define _overloadoutl(_1, _2, _3, _4, name, ...) name
#define outl1(a) cout<<a<<endl
#define outl2(a, b) cout<<a<<endl<<b<<endl
#define outl3(a, b, c) cout<<a<<endl<<b<<endl<<c<<endl
#define outl4(a, b, c, d) cout<<a<<endl<<b<<endl<<c<<endl<<d<<endl
#define outl(...) _overloadout(__VA_ARGS__,outl4,outl3,outl2,outl1)(__VA_ARGS__)

#define na(a, n) a.resize(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 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 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 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) 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, p2;
void setmod() {
    fac.resize(mint_len);
    finv.resize(mint_len);
    inv.resize(mint_len);
    p2.resize(mint_len);
    inv[1] = fac[0] = finv[0] = p2[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, p2[i] = p2[i - 1] * 2 % 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;
    int 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 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 count(V &a, const T k) { return a == k; }
template<typename V, typename T> int count(vector<V> &a, const T k) {
    int ret = 0;
    fora(v, a)ret += count(v, k);
    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 n) {
    vi ve(n);
    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> inline void rev(vector<T> &a) {
    reverse(all(a));
}
void inline rev(string &a) {
    reverse(all(a));
}

int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }
int keta(int v) {
    int cou = 0;
    while (v) { cou++, v %= 10; }
    return cou;
}
int dsum(int v) {
    int ret = 0;
    for (; v; v /= 10)ret += v % 10;
    return ret;
}
inline int sumd(int v) { return dsum(v); }
//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, 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) { while (q.size())q.pop(); }
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;
}
//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;
}
//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);
}
//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;
}
inline ll bit(int n) { return (1LL << (n)); }
inline ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }
//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);
}
//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() );

void 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();
    }
}

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];
}
inline int gcd(int a, int b) {
    while (b) a %= b, swap(a, b);
    return a;
}
inline int gcd(vi b) {
    ll res = b[0];
    for (auto &&v :b)res = gcd(v, res);
    return res;
}
inline ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
inline 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 = 1e7;
    vector<int> primes;
    void setprime(int size = 1e7) {
        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);
            }
        }
        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 *= 2;

        bestt.resize(2 * sn - 1);
        rep(i, k)bestt[i + sn - 1] = i;//時間を持つ
        rer(i, sn - 2) {
            int lt = i * 2 + 1, rt = i * 2 + 2;
            if (dep[bestt[i * 2 + 1]] < dep[bestt[i * 2 + 2]]) {
                bestt[i] = bestt[i * 2 + 1];
            } else {
                bestt[i] = bestt[i * 2 + 2];
            }
        }
    }
    //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, 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<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);
}

int n, m, k, d, H, W, x, y, z, q;
int cou;
vi a, b, c;
//vvi (s, 0, 0);
vvc (ba, 0, 0);
vp p;
str s;
//0 index
template<typename T = int> class BIT {
public:
    int n;
    vector<T> dat;
    BIT(int n) : n(n) { dat.assign(n, 0); }
    // [0, k)
    inline T sum(int k) {
        if (k >= n) k = n - 1;
        T ret = 0;
        for (int x = k - 1; x >= 0; x = (x & (x + 1)) - 1) {
            ret += dat[x];
        }
        return ret;
    }
    // [l, r)
    inline T sum(int l, int r) { return sum(r) - sum(l); }
    inline T get(int l, int r) { return sum(r) - sum(l); }
    inline T get(int k) { return sum(k); }
    inline T operator[](int k) {
        assert(0 <= k && k < n);
        return sum(k + 1) - sum(k);
    }
    inline T operator()(int k) { return sum(k); }
    inline T operator()(int l, int r) { return sum(l, r); }
    //0 index
    inline void add(int k, T val = 1) { for (int x = k; x < n; x |= x + 1)dat[x] += val; }
    inline void update(int k, T val = 0) { add(k, -operator[](k) + val); }
    inline void del(int k) { update(k, 0); }
    inline void clear() { fill(dat, 0); }
    inline void debu() {
#ifdef _DEBUG
        vi res;
        rep(i, min(10, n)) {
            res.pb(operator[](i));
        }
        deb(res);
#endif
        ;
    }
    //合計がw以上となる最小のxを返す
    inline int lower_bound(int w) {
        if (w <= 0) return -1;
        int x = 0;
        int k = 1;
        while (k * 2 <= n) k *= 2;
        for (; k > 0; k /= 2) {
            if (x + k <= n && dat[x + k - 1] < w) {
                w -= dat[x + k - 1];
                x += k;
            }
        }
        return x;
    }
};
template<class T> int couinv(const vector<T> &a) {
    vector<T> b = a;
    compress(b);
    int n = sz(b);
    BIT<int> bt(n + 3);
    int cou = 0;
    rer(i, n - 1) {
        cou += bt.get(a[i]);
        bt.add(a[i], 1);
    }
    return cou;
}

bool calc(int v) {
    b.clear();
    BIT<> bt(2 * k5);
    rep(i, n)b.pb(a[i] >= v ? 1 : -1);
    int sum = 0, res = 0;
    bt.add(k5);
    rep(i, n) {
        sum += b[i];
        res += bt.sum(sum + k5 + 1);
        bt.add(sum + k5);
    }
    return res * 2 >= (n + 1) * n / 2;
}
signed main() {
    cin >> n;
    na(a, n);
    int ok = 1, ng = inf;
    while (abs(ok - ng) > 1) {
        int mid = (ok + ng) / 2;
        if (calc(mid))ok = mid;
        else ng = mid;
    }
    cout << ok << endl;
    return 0;
}