#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using db = double;
using str = string;
using pi = pair<int,int>;
using pl = pair<ll,ll>;
using pd = pair<db,db>;
using vi = vector<int>;
using vb = vector<bool>;
using vl = vector<ll>;
using vd = vector<db>;
using vs = vector<str>;
using vpi = vector<pi>;
using vpl = vector<pl>;
using vpd = vector<pd>;
#define tcT template<class T
#define tcTU tcT, class U
tcT> using V = vector<T>;
tcT, size_t SZ> using AR = array<T,SZ>;
tcT> using PR = pair<T,T>;
// pairs
// #define mp make_pair
#define f first
#define s second
// vectors
// size(x), rbegin(x), rend(x) need C++17
#define sz(x) int((x).size())
#define bg(x) begin(x)
#define all(x) bg(x), end(x)
#define rall(x) x.rbegin(), x.rend()
#define sor(x) sort(all(x))
#define rsz resize
#define ins insert
#define ft front()
#define bk back()
#define pb push_back
#define eb emplace_back
#define pf push_front
// #define rtn return
#define lb lower_bound
#define ub upper_bound
/* Some Codes Skipped */
// loops
#define FOR(i,a,b) for (int (i) = (a); (i) < (b); ++(i))
#define REP(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int (i) = (b)-1; (i) >= (a); --(i))
#define PER(i,a) ROF(i,0,a)
#define rep(a) REP(_,a)
#define each(a,x) for (auto& a: x)
const int MOD = 1e9+7;
const int MX = 2e5+5;
const ll INF = 1e18;
const db PI = acos((db)-1);
const int dx[4] = {1,0,-1,0}, dy[4] = {0,1,0,-1}; // for every grid problem
tcT> bool ckmin(T& a, const T& b) {
return b < a ? a = b, 1 : 0; } // set a = min(a,b)
tcT> bool ckmax(T& a, const T& b) {
return a < b ? a = b, 1 : 0; }
tcT> void remDup(vector<T>& v) { // sort and remove duplicates
sort(all(v)); v.erase(unique(all(v)),end(v)); }
// Some Codes Skipped
inline namespace FileIO {
void setIn(str s) {freopen(s.c_str(),"r",stdin);}
void setOut(str s) {freopen(s.c_str(),"w",stdout);}
void setIO(str s = "") {
cin.tie(0)->sync_with_stdio(0); // unsync C / C++ I/O streams
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) setIn(s+".in"), setOut(s+".out"); // for old USACO
}
}
int main() {
setIO();
// you should actually read the stuff at the bottom
}
/* stuff you should look for
* int overflow, array bounds
* special cases (n=1?)
* do smth instead of nothing and stay organized
* WRITE STUFF DOWN
* DO NOT GET STUCK ON ONE APPROACH
*/