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#include <bits/stdc++.h>
#define MP make_pair
#define PB push_back
#define int long long
#define st first
#define nd second
#define rd third
#define FOR(i, a, b) for(int i =(a); i <=(b); ++i)
#define RE(i, n) FOR(i, 1, n)
#define FORD(i, a, b) for(int i = (a); i >= (b); --i)
#define REP(i, n) for(int i = 0;i <(n); ++i)
#define VAR(v, i) __typeof(i) v=(i)
#define FORE(i, c) for(VAR(i, (c).begin()); i != (c).end(); ++i)
#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((int)(x).size())
#ifdef LOCAL
#define debug(x) {cerr <<#x<<" = " <<x<<"\n"; }
#define debug2(x, y) {cerr <<#x<<" = " <<x<<", "<<#y <<" = " <<y <<"\n";}
#define debug3(x, y, z) {cerr <<#x<<" = " <<x<<", "<<#y <<" = " <<y <<", "<<#z<<" = "<<z<<"\n";}
#define debug4(x, y, z, t) {cerr <<#x<<" = " <<x<<", "<<#y <<" = " <<y <<", "<<#z<<" = "<<z<<", "<<#t <<" = " <<t<<"\n";}
#define debugv(x) {{cerr <<#x <<" = "; FORE(itt, (x)) cerr <<*itt <<", "; cerr <<"\n"; }}
#else
#define debug(x)
#define debug2(x, y)
#define debug3(x, y, z)
#define debug4(x,y,z,t)
#define debugv(x)
#define cerr if(0)cout
#endif
#define make(type, x) type x; cin>>x;
#define make2(type, x, y) type x, y; cin>>x>>y;
#define make3(type, x, y, z) type x, y, z; cin>>x>>y>>z;
#define make4(type, x, y, z, t) type x, y, z, t; cin>>x>>y>>z>>t;
#define next ____next
#define prev ____prev
#define left ____left
#define hash ____hash
using namespace std;
typedef long long ll;
typedef long double LD;
typedef pair<int, int> PII;
typedef pair<ll, ll> PLL;
typedef vector<int> VI;
typedef vector<VI> VVI;
typedef vector<ll> VLL;
typedef vector<pair<int, int> > VPII;
typedef vector<pair<ll, ll> > VPLL;
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