#include<bits/stdc++.h>
using namespace std;

using ll = int64_t;
using ull = uint64_t;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;

#define lowbit(x) ((x) & -(x))
#define popcount(x) (__builtin_popcountll(ll(x)))
#define parity(x) (__builtin_parityll(ll(x)))
#define msp(x) (63LL - __builtin_clzll(ll(x)))
#define lsp(x) (__builtin_ctzll(ll(x)))

#define upair ull
#define umake(x, y) (ull(x) << 32 | ull(y))
#define u1(p) ((p) >> 32)
#define u2(p) ((p) & ((1ULL << 32) - 1))
#define ult std::less<upair>
#define ugt std::greater<upair>

#define ipair ull
#define imake(x, y) (umake(x, y))
#define i1(p) (int(u1(p)))
#define i2(p) (int(u2(p)))
struct ilt {
    bool operator()(const ipair& a, const ipair& b) const {
        if (i1(a) == i1(b)) return i2(a) < i2(b);
        else return i1(a) < i1(b);
    }
};
struct igt {
    bool operator()(const ipair& a, const ipair& b) const {
        if (i1(a) == i1(b)) return i2(a) > i2(b);
        else return i1(a) > i1(b);
    }
};

#define read(x, type) type x; cin >> x;
#define readvec(a, n, type) vector<type> a(n); for (int i = 0; i < (n); ++i) cin >> a[i];
#define putvec(a) for (auto&& x : a) cout << x << ' '; cout << endl;
#define debug(x) cerr << #x" = " << x << endl;

#define pa(a) __typeof(a) pa; pa.push_back({}); for (auto&&x : a) pa.push_back(pa.back() + x);
#define sa(a) __typeof(a) sa(a.size() + 1); {int n = a.size(); for (int i = n - 1; i >= 0; --i) sa[i] = sa[i + 1] + a[i];};