#include "hack.h"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
static bool has_collision(const vector<ll>& a, const vector<ll>& b) {
vector<ll> q;
q.reserve(a.size() + b.size());
for (ll x : a) q.push_back(x);
for (ll x : b) q.push_back(x);
return collisions(q) > 0;
}
int hack() {
const ll P = 14009;
const ll L = 500000000LL;
const ll R = 1000000000LL;
// A = {1, 2, ..., P}
vector<ll> A(P);
iota(A.begin(), A.end(), 1);
// B = consecutive multiples of P covering [L, R]
vector<ll> B;
ll start = (L / P) * P;
for (ll x = start; x <= R + P; x += P) {
B.push_back(x);
}
vector<ll> V = A, U = B;
// Invariant: V ∪ U contains at least one colliding pair
while (V.size() > 1 || U.size() > 1) {
if (V.size() < U.size()) swap(V, U);
size_t mid = V.size() / 2;
vector<ll> right(V.begin() + mid, V.end());
if (has_collision(right, U)) {
V = move(right);
} else {
V.resize(mid);
}
}
ll X = llabs(V[0] - U[0]); // X is a positive multiple of n
// Remove redundant prime factors from X until it becomes exactly n
ll ans = X;
ll tmp = X;
for (ll p = 2; p * p <= tmp; ++p) {
if (tmp % p != 0) continue;
while (tmp % p == 0) tmp /= p;
while (ans % p == 0) {
ll cand = ans / p;
if (collisions({1, cand + 1}) > 0) {
ans = cand; // still divisible by n
} else {
break;
}
}
}
if (tmp > 1) {
while (ans % tmp == 0) {
ll cand = ans / tmp;
if (collisions({1, cand + 1}) > 0) {
ans = cand;
} else {
break;
}
}
}
return (int)ans;
}
