#include<bits/stdc++.h>
#define FOR(i, a, b) for (int i = (a), _b = (b); i <= _b; i++)
#define FORD(i, b, a) for (int i = (b), _a = (a); i >= _a; i--)
#define REP(i, n) for (int i = 0, _n = (n); i < _n; i++)
#define FORE(i, v) for (__typeof((v).begin()) i = (v).begin(); i != (v).end(); i++)
#define ALL(v) (v).begin(), (v).end()
#define IS_INF(x)   (std::isinf(x))
#define IS_NAN(x)   (std::isnan(x))
#define fi   first
#define se   second
#define MASK(i) (1LL << (i))
#define BIT(x, i) (((x) >> (i)) & 1)
#define div   ___div
#define next   ___next
#define prev   ___prev
#define left   ___left
#define right   ___right
#define __builtin_popcount __builtin_popcountll

#define ios::sync_with_stdio(false),cin.tie(0)
#define endl "\n"
#define LL long long 
#define PII pair<int,int>