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/**
* Youth gives you light please don‘t let it down.
* ▄▄▄▄▄
*            ???██████▄▄▄       _______________
*          ▄▄▄▄▄  █████████▄  /                 *         ????█████▌ ??▄ ??█ | Gotta go fast!   |
*       ??█████▄▄ ?██████▄██ | _________________/
*       ?▄▄▄▄▄  ??█▄?█════█? |/
*            ???▄  ??███ ?       ▄▄
*         ▄███??██▄████████▄ ▄??????█▌   ______________________________ 
*       ██?▄▄▄██?▄███? ??████      ▄██  █                              \\ 
*    ▄???▄██▄??▌████??????███     ▌▄▄????█_____________________________//
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* \  ???????????????????????????????????? ? 
* \- ▌          GNU C++17 7.3.0            ? ?      
*  - ▌                                         ? 
* /- ▌            Go Go Go !               ? ?           
* /  ???????????????????????????????????? ?       
*               ██
**/
#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<bitset>
#include<cassert>
#include<cctype>
#include<cmath>
#include<cstdlib>
#include<ctime>
#include<deque>
#include<iomanip>
#include<list>
#include<map>
#include<queue>
#include<set>
#include<stack>
#include<vector>
using namespace std;
typedef long long ll;
typedef long double ld;
#define ios ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define mem(a, b) memset(a, b, sizeof(a))
const double pi = acos(-1.0);
const double eps = 1e-6;
const ll mod = 1e9 + 7;
const int inf = 0x3f3f3f3f;
const ll INF = 0x3f3f3f3f3f3f3f3f;
const int maxn = 1e7 + 5;