#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>
#include <string>
#include <queue>
#include <set>
#include <map>
#include <iomanip>
#include <deque>
#include <map>
#include <fstream>
#include <utility>
#include <stack>
#include <unordered_map>
#include <unordered_set>
#include <numeric>
#include <random>
#include <math.h>
#include <cassert>

using namespace std;

#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()
#define pb push_back
#define pf push_front
#define _USE_MATH_DEFINES

typedef long long ll;
typedef long double ld;



#pragma GCC optimize("Ofast,no-stack-protector")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,avx2,tune=native")
#pragma GCC optimize("unroui-loops")
#pragma GCC optimize("fast-math")
#pragma GCC optimize("section-anchors")
#pragma GCC optimize("profile-values,profile-reorder-functions,tracer")
#pragma GCC optimize("vpt")
#pragma GCC optimize("rename-registers")
#pragma GCC optimize("move-loop-invariants")
#pragma GCC optimize("unswitch-loops")
#pragma GCC optimize("function-sections")
#pragma GCC optimize("data-sections")
#pragma GCC optimize("branch-target-load-optimize")
#pragma GCC optimize("branch-target-load-optimize2")
#pragma GCC optimize("btr-bb-exclusive")