#include <bits/stdc++.h>
using namespace std
#define fastcin ios_base::sync_with_stdio(false)
#define precisecout cout << setprecision(999)
#define pb push_back
#define inf 0x3f3f3f3f
#define mod109 1000000007
#define ll long long
#define binary(x) bitset<8*sizeof(x)>(x)