#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned int ul;
typedef unsigned long long ull;
#define MASK(i) (1ULL << (i))
#define GETBIT(mask, i) (((mask) >> (i)) & 1)
#define ALL(v) (v).begin(), (v).end()
ll max(ll a, ll b){return (a > b) ? a : b;}
ll min(ll a, ll b){return (a < b) ? a : b;}
ll gcd(ll a, ll b){return __gcd(abs(a), abs(b));}
ll lcm(ll a, ll b){return abs(a) / gcd(a, b) * abs(b);}
ll LASTBIT(ll mask){return (mask) & (-mask);}
int pop_cnt(ull mask){return __builtin_popcountll(mask);}
int ctz(ull mask){return __builtin_ctzll(mask);}
int logOf(ull mask){return 63 - __builtin_clzll(mask);}
mt19937_64 rng(chrono::high_resolution_clock::now().time_since_epoch().count());
ll rngesus(ll l, ll r){return l + (ull) rng() % (r - l + 1);}
template <class T1, class T2>
bool maximize(T1 &a, T2 b){
if (a < b) {a = b; return true;}
return false;
}
template <class T1, class T2>
bool minimize(T1 &a, T2 b){
if (a > b) {a = b; return true;}
return false;
}
template <class T>
void printArr(T container, string separator = " ", string finish = "\n", ostream &out = cout){
for(auto item: container) out << item << separator;
out << finish;
}
template <class T>
void remove_dup(vector<T> &a){
sort(ALL(a));
a.resize(unique(ALL(a)) - a.begin());
}
#include "hack.h"
int get_collision(vector<ll> x){
for(auto &i: x) i++;
return collisions(x);
}
int find_dih(int x){
vector<int> divisor;
for(int i = 1; i * i <= x; ++i) {
if (x % i == 0){
divisor.push_back(i);
divisor.push_back(x / i);
}
}
remove_dup(divisor);
for(int i: divisor) if (get_collision({0, i})) return i;
return -1;
}
vector<ll> make_arr(int s, int len, int step){
vector<ll> cur;
for(int i = 0; i < len; ++i) cur.push_back(s + i * step);
return cur;
}
int find_lesser_mod(){
const int MOD = 179;
vector<ll> first_shot;
for(int i = 0; i < MOD; ++i) first_shot.push_back(i);
for(int i = 1; i < MOD; ++i) first_shot.push_back(i * MOD);
if (get_collision(first_shot) == 0) return -1;
int l = 1, r = MOD * MOD;
while(l < r){
int mid = (l + r) >> 1;
vector<ll> cur; cur.push_back(0);
for(int i = l; i <= mid; ++i) cur.push_back(i);
if (get_collision(cur)) r = mid;
else l = mid + 1;
}
return find_dih(l);
}
vector<ll> express_range(int l, int r){
int s = sqrt(r - l + 1);
vector<ll> ans;
for(int i = 0; i < s; ++i) ans.push_back(i);
for(int i = r; i >= l; i -= s){
ans.push_back(i);
if (i - (s - 1) <= l) break;
}
return ans;
}
int hack(){
int less_mod = find_lesser_mod();
if (less_mod != -1) return less_mod;
int l = 5e8 + 1, r = 1e9;
while(l < r){
int mid = (l + r) >> 1;
vector<ll> S = express_range(l, mid);
if (get_collision(S)) r = mid;
else l = mid + 1;
}
return find_dih(l);
}
