#include "hack.h"
#include <vector>
using namespace std;
#ifndef t9unkubj
#define dbg(...) 199958
#else
#define dbg(...) debug_func(0, #__VA_ARGS__, __VA_ARGS__)
#endif
#pragma GCC optimize("O3")
#include"bits/stdc++.h"
using namespace std;
using uint=unsigned;
using ll=long long;
using ull=unsigned long long;
using ld=long double;
using i128=__int128;
template<class T>using vc=vector<T>;
template<class T>using vvc=vc<vc<T>>;
template<class T>using vvvc=vc<vvc<T>>;
template<class T>using vvvvc=vc<vvvc<T>>;
template<class T>using smpq=priority_queue<T,vc<T>,greater<T>>;
template<class T>using bipq=priority_queue<T>;
using vi=vc<int>;
using vvi=vvc<int>;
using vl=vc<ll>;
using vvl=vvc<ll>;
#define rep(i,n) for(ll i=0;i<(ll)(n);i++)
#define REP(i,j,n) for(ll i=(j);i<(ll)(n);i++)
#define DREP(i,n,m) for(ll i=(n);i>=(m);i--)
#define drep(i,n) for(ll i=(ll(n)-1);i>=0;i--)
#define all(x) x.begin(),x.end()
#define rall(x) x.rbegin(),x.rend()
#define pb push_back
template<class T,class F>
bool chmin(T &x, F y){
if(x>y){
x=y;
return true;
}
return false;
}
template<class T, class F>
bool chmax(T &x, F y){
if(x<y){
x=y;
return true;
}
return false;
}
template<class T>
T sum(const vector<T>&v){
return accumulate(all(v),T(0));
}
template<class T>
T min(const vector<T>&v){
return *min_element(all(v));
}
template<class T>
T max(const vector<T>&v){
return *max_element(all(v));
}
void YesNo(bool y){
cout<<(y?"Yes":"No")<<endl;
}
void Yes(){
cout<<"Yes"<<endl;
}
void No(){
cout<<"No"<<endl;
}
template<class T>
void unique(vc<T>&a){
a.erase(unique(all(a)),a.end());
}
vvi readgraph(int n,int m,int off = -1){
vvi g(n);
rep(i, m){
int u,v;
cin>>u>>v;
u+=off,v+=off;
g[u].push_back(v);
g[v].push_back(u);
}
return g;
}
vvi readtree(int n,int off=-1){
return readgraph(n,n-1,off);
}
template<class T>
vc<T> presum(vc<T> &a){
vc<T> ret(a.size()+1);
rep(i,a.size())ret[i+1]=ret[i]+a[i];
return ret;
}
template<class T, class F>
vc<T> &operator+=(vc<T> &a,F b){
for (auto&v:a)v += b;
return a;
}
template<class T, class F>
vc<T> &operator-=(vc<T>&a,F b){
for (auto&v:a)v-=b;
return a;
}
ostream&operator<<(ostream&os,i128 num) {
if(num==0){
os<<"0";
return os;
}
if(num<0){
os<<"-";
num=-num;
}
string res;
while(num){
res+='0'+static_cast<int>(num%10);
num/=10;
}
reverse(all(res));
os<<res;
return os;
}
istream&operator>>(istream& is,i128&num) {
string s;
is>>s;
bool neg=0;
num=0;
if(s[0]=='-'){
neg=1;
s=s.substr(1);
}
for(auto&c:s){
if (c>='0'&&c<='9'){
num=num*10+(c-'0');
}else{
is.setstate(ios::failbit);
return is;
}
}
if(neg){
num=-num;
}
return is;
}
void scan(int&a) { cin >> a; }
void scan(ll&a) { cin >> a; }
void scan(string&a) { cin >> a; }
void scan(char&a) { cin >> a; }
void scan(uint&a) { cin >> a; }
void scan(ull&a) { cin >> a; }
void scan(bool&a) { cin >> a; }
void scan(ld&a){ cin>> a; }
void scan(i128&a){ cin>> a; }
template<class T> void scan(vector<T>&a) { for(auto&x:a) scan(x); }
void read() {}
template<class Head, class... Tail> void read(Head&head, Tail&... tail) { scan(head); read(tail...); }
#define INT(...) int __VA_ARGS__; read(__VA_ARGS__);
#define LL(...) ll __VA_ARGS__; read(__VA_ARGS__);
#define ULL(...) ull __VA_ARGS__; read(__VA_ARGS__);
#define STR(...) string __VA_ARGS__; read(__VA_ARGS__);
#define CHR(...) char __VA_ARGS__; read(__VA_ARGS__);
#define DBL(...) double __VA_ARGS__; read(__VA_ARGS__);
#define LD(...) ld __VA_ARGS__; read(__VA_ARGS__);
#define I128(...) i128 __VA_ARGS__; read(__VA_ARGS__);
#define VC(type, name, ...) vector<type> name(__VA_ARGS__); read(name);
#define VVC(type, name, size, ...) vector<vector<type>> name(size, vector<type>(__VA_ARGS__)); read(name);
void print(int a) { cout << a; }
void print(ll a) { cout << a; }
void print(string a) { cout << a; }
void print(char a) { cout << a; }
void print(uint a) { cout << a; }
void print(bool a) { cout << a; }
void print(ull a) { cout << a; }
void print(double a) { cout << a; }
void print(ld a){ cout<< a; }
void print(i128 a){ cout<< a; }
template<class T> void print(vector<T>a) { for(int i=0;i<(int)a.size();i++){if(i)cout<<" ";print(a[i]);}cout<<endl;}
void PRT() { cout <<endl; return ; }
template<class T> void PRT(T a) { print(a); cout <<endl; return; }
template<class Head, class... Tail> void PRT(Head head, Tail ... tail) { print(head); cout << " "; PRT(tail...); return; }
struct ioset{
ioset(){
cin.tie(0)->sync_with_stdio(0);
#ifdef t9unkubj
cout<<fixed<<setprecision(6);
#else
cout<<fixed<<setprecision(20);
#endif
}
}ioset_______;
struct MY_TIMER{
double start_time;
MY_TIMER(){
start_time=clock();
}
double out(){
return (clock()-start_time)/double(CLOCKS_PER_SEC);
}
}TIMER;
int B=8e3;
int hack(){
ll tar=0;
int l=1,r=1e9;
while(r-l>100){
int B=sqrt(r-l);
vc<ll>first;
REP(i,1,B+1)first.push_back(i);
//[1,ac) までにvalidなものが存在する
int l0=max(1,l/B),r0=(r+B-1)/B;
if(l0+2>r0)break;
int mid=l0+r0>>1;
REP(i,max(2,l0),mid)first.push_back(i*B);
if(collisions(first))r=(mid-1)*B;
else l=max(1,(mid-1)*B);
dbg(l,r);
}
dbg(l,r);
for(int i=max(2,l);;i++){
if(collisions({(int)1,i})){
return i-1;
}
}
}
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