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/*
                                             __----~~~~~~~~~~~------___
                                  .  .   ~~//====......          __--~ ~~
                  -.            \_|//     |||\\  ~~~~~~::::... /~
               ___-==_       _-~o~  \/    |||  \\            _/~~-
       __---~~~.==~||\=_    -_--~/_-~|-   |\\   \\        _/~
   _-~~     .=~    |  \\-_    '-~7  /-   /  ||    \      /
 .~       .~       |   \\ -_    /  /-   /   ||      \   /
/  ____  /         |     \\ ~-_/  /|- _/   .||       \ /
|~~    ~~|--~~~~--_ \     ~==-/   | \~--===~~        .\
         '         ~-|      /|    |-~\~~       __--~~
                     |-~~-_/ |    |   ~\_   _-~            /\
                          /  \     \__   \/~                \__
                      _--~ _/ | .-~~____--~-/                  ~~==.
                     ((->/~   '.|||' -_|    ~~-/ ,              . _||
                                -_     ~\      ~~---l__i__i__i--~~_/
                                _-~-__   ~)  \--______________--~~
                              //.-~~~-~_--~- |-------~~~~~~~~
                                     //.-~~~--\
*/
#define _CRT_SECURE_NO_WARNINGS
#include<cstdio>
#include<iostream>
#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>
#define lt k << 1
#define rt k << 1 | 1
#define lowbit(x) x & (-x)
#define lson l, mid, lt
#define rson mid+1, r, rt
using namespace std;
typedef long long ll;
typedef long double ld;
#define fin freopen("in.txt", "r", stdin);
#define fout freopen("out.txt", "w", stdout);
#define ios ios::sync_with_stdio(false); cin.tie(0); cout.tie(0);
#define mem(a, b) memset(a, b, sizeof(a))
//#define int ll
const double pi = acos(-1.0);
const double eps = 1e-8;
const ll mod = 1e9 + 7;
const int inf = 0x3f3f3f3f;
const ll INF = 0x3f3f3f3f3f3f3f3f;
const int maxn = 2e5 + 500;