#include<bits/allocator.h>
#pragma GCC optimize("Ofast,unroll-loops")
#pragma GCC target("avx2,fma,bmi,bmi2,popcnt,lzcnt,tune=native")
#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define ull unsigned long long
#define int128 __int128_t
#define double long double
#define gcd __gcd
#define lcm(a, b) ((a)/gcd(a, b)*(b))
#define sqrt sqrtl
#define log2 log2l
#define log10 log10l
#define floor floorl
#define to_string str
#define yes cout << "YES"
#define no cout << "NO"
#define trav(i, a) for (auto &i: (a))
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define sz(a) (int)a.size()
#define Max(a) *max_element(all(a))
#define Min(a) *min_element(all(a))
#define Find(a, n) (find(all(a), n) - a.begin())
#define Count(a, n) count(all(a), n)
#define Upper(a, n) (upper_bound(all(a), n) - a.begin())
#define Lower(a, n) (lower_bound(all(a), n) - a.begin())
#define next_perm(a) next_permutation(all(a))
#define prev_perm(a) prev_permutation(all(a))
#define sorted(a) is_sorted(all(a))
#define sum(a) accumulate(all(a), 0)
#define sumll(a) accumulate(all(a), 0ll)
#define Sort(a) sort(all(a))
#define Reverse(a) reverse(all(a))
#define Unique(a) Sort(a), (a).resize(unique(all(a)) - a.begin())
#define pb push_back
#define eb emplace_back
#define popcount __builtin_popcount
#define popcountll __builtin_popcountll
#define clz __builtin_clz
#define clzll __buitlin_clzll
#define ctz __builtin_ctz
#define ctzll __builtin_ctzll
#define open(s) freopen(s, "r", stdin)
#define write(s) freopen(s, "w", stdout)
#define fileopen(s) open((string(s) + ".inp").c_str()), write((string(s) + ".out").c_str());
#define For(i, a, b) for (auto i = (a); i < (b); ++i)
#define Fore(i, a, b) for (auto i = (a); i >= (b); --i)
#define FOR(i, a, b) for (auto i = (a); i <= (b); ++i)
#define ret(s) return void(cout << s);
constexpr int mod = 1e9 + 7, mod2 = 998244353;
constexpr double eps = 1e-9;
const double PI = acos(-1);
constexpr ull npos = string::npos;
constexpr int dx[] = {1, 0, -1, 0, 1, 1, -1, -1}, dy[] = {0, 1, 0, -1, 1, -1, 1, -1};
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using cd = complex<double>;
mt19937 mt(chrono::system_clock::now().time_since_epoch().count());
mt19937_64 mt64(chrono::system_clock::now().time_since_epoch().count());
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<ll> vll;
typedef vector<vll> vvll;
typedef vector<double> vdo;
typedef vector<vdo> vvdo;
typedef vector<string> vs;
typedef vector<pii> vpair;
typedef vector<vpair> vvpair;
typedef vector<bool> vb;
typedef vector<vb> vvb;
typedef vector<char> vc;
typedef vector<vc> vvc;
typedef vector<cd> vcd;
typedef priority_queue<int> pq;
typedef priority_queue<int, vi, greater<int>> pqg;
typedef priority_queue<ll> pqll;
typedef priority_queue<ll, vll, greater<ll>> pqgll;
ll add(ll a, ll b, int m) {a = (a >= m ? a % m: a);b = (b >= m ? b % m: b);a+=b;return a >= m ? a - m: a;}
ll sub(ll a, ll b, int m) {a = (a >= m ? a % m: a);b = (b >= m ? b % m: b);a-=b;return a < 0 ? a + m: a;}
ll mul(ll a, ll b, int m) {a = (a >= m ? a % m: a);b = (b >= m ? b % m: b);return a*b % m;}
ll bin_mul(ll a, ll b, ll m) {a = (a >= m ? a % m: a);b = (b >= m ? b % m: b);ll x = 0;while (b) {if (b & 1) x = (x + a) % m;a = (a + a) % m;b>>=1;}return x;}
ll bin_pow(ll a, ll b, ll m) {ll x = 1;a = (a >= m ? a % m: a); while (b) {if (b & 1) x = bin_mul(x, a, m);a = bin_mul(a, a, m);b>>=1;}return x;}
ll power(ll a, ll b, int m) {ll x = 1;a = (a >= m ? a % m: a); while (b) {if (b & 1) x = x*a % m;a = a*a % m;b>>=1;}return x;}
ll power(ll a, ll b) {ll x = 1;while (b) {if (b & 1) x = x*a;a = a*a;b>>=1;}return x;}
ll ceil(ll a, ll b) {return (a + b - 1)/b;}
ll to_int(const string &s) {ll x = 0; for (int i = (s[0] == '-'); i < sz(s); ++i) x = x*10 + s[i] - '0';return x*(s[0] == '-' ? -1: 1);}
bool is_prime(ll n) {if (n < 2) return 0;if (n < 4) return 1;if (n % 2 == 0 || n % 3 == 0) return 0;for (ll i = 5; i*i <= n; i+=6) {if(n % i == 0 || n % (i + 2) == 0) return 0;}return 1;}
bool is_square(ll n) {ll k = sqrt(n); return k*k == n;}
ll factorial(int n) {ll x = 1;for (int i = 2; i <= n; ++i) x*=i;return x;}
ll factorial(int n, int m) {ll x = 1;for (ll i = 2; i <= n; ++i) x = x*i % m;return x;}
bool is_power(ll n, ll k) {while (n % k == 0) n/=k;return n == 1ll;}
string str(ll n) {if (n == 0) return "0"; string s = ""; bool c = 0; if (n < 0) c = 1, n = -n; while (n) {s+=n % 10 + '0'; n/=10;} if (c) s+='-'; Reverse(s); return s;}
string repeat(const string &s, int n) {if (n < 0) return ""; string x = ""; while (n--) x+=s; return x;}
string bin(ll n) {string s = ""; while (n) {s+=(n & 1) + '0'; n>>=1;} Reverse(s); return s;}
void sieve(vector<bool> &a) {int n = a.size(); a[0] = a[1] = 0; for (int i = 4; i < n; i+=2) a[i] = 0; for (int i = 3; i*i < n; i+=2) {if (a[i]) {for (int j = i*i; j < n; j+=(i << 1)) a[j] = 0;}}}
void sieve(bool a[], int n) {a[0] = a[1] = 0; for (int i = 4; i < n; i+=2) a[i] = 0; for (int i = 3; i*i < n; i+=2) {if (a[i]) {for (int j = i*i; j < n; j+=(i << 1)) a[j] = 0;}}}
void sieve(vector<int> &a) {int n = a.size(); for (int i = 2; i < n; i+=2) a[i] = 2; for (int i = 3; i*i < n; i+=2) {if (!a[i]) {for (int j = i; j < n; j+=(i << 1)) a[j] = i;}} for (int i = 3; i < n; i+=2) {if (!a[i]) a[i] = i;}}
void sieve(int a[], int n) {for (int i = 2; i < n; i+=2) a[i] = 2; for (int i = 3; i*i < n; i+=2) {if (!a[i]) {for (int j = i; j < n; j+=(i << 1)) a[j] = i;}} for (int i = 3; i < n; i+=2) {if (!a[i]) a[i] = i;}}
vector<pii> factorize(int n) {vector<pii> a; for (int i = 2; i*i <= n; ++i) {if (n % i == 0) {int k = 0; while (n % i == 0) ++k, n/=i; a.emplace_back(i, k);}} if (n > 1) a.emplace_back(n, 1); return a;}
int rand(int l, int r) {return uniform_int_distribution<int>(l, r)(mt);}
ll rand(ll l, ll r) {return uniform_int_distribution<ll>(l, r)(mt64);}
int Log2(int n) {return 31 - __builtin_clz(n);}
ll Log2(ll n) {return 63 - __builtin_clzll(n);}
template<class T> void compress(vector<T> &a) {vector<T> b; for (T &i: a) b.push_back(i); sort(all(b)); b.resize(unique(all(b)) - b.begin()); for (T &i: a) i = lower_bound(all(b), i) - b.begin() + 1;}
template<class A, class B> bool ckmin(A &a, const B &b) {return a > b ? a = b, 1: 0;}
template<class A, class B> bool ckmax(A &a, const B &b) {return a < b ? a = b, 1: 0;}
template<class A, class B> istream& operator>>(istream& in, pair<A, B> &p) {in >> p.first >> p.second; return in;}
template<class A, class B> ostream& operator<<(ostream& out, const pair<A, B> &p) {out << p.first << ' ' << p.second; return out;}
template<class T> istream& operator>>(istream& in, vector<T> &a) {for (auto &i: a) in >> i; return in;}
template<class T> ostream& operator<<(ostream& out, const vector<T> &a) {for (auto &i: a) out << i << ' '; return out;}
template<class T> istream& operator>>(istream& in, vector<vector<T>> &a) {for (auto &i: a) in >> i; return in;}
template<class T> ostream& operator<<(ostream& out, const vector<vector<T>> &a) {for (auto &i: a) out << i << '\n'; return out;}
template<class T> istream& operator>>(istream& in, deque<T> &a) {for (auto &i: a) in >> i; return in;}
template<class T> ostream& operator<<(ostream& out, const deque<T> &a) {for (auto &i: a) out << i << ' '; return out;}
// istream& operator>>(istream& in, __int128_t &a) {string s; in >> s; a = 0; for (auto &i: s) a = a*10 + (i - '0'); return in;}
// ostream& operator<<(ostream& out, __int128_t a) {string s = ""; if (a < 0) s+='-', a = -a; if (a == 0) s+='0'; while (a > 0) {s+=(int)(a % 10) + '0'; a/=10;} Reverse(s); out << s; return out;}
#include "game.h"
constexpr int N = 1e9;
struct node {
int l, r;
ll x;
node *lt, *rt;
node(int l = 0, int r = 0, ll x = 0): l(l), r(r), x(x) {
lt = rt = nullptr;
}
void upd(int i, ll v) {
if (l == r) return void(x = v);
int m = (l + r) >> 1;
node *&t = i <= m ? lt: rt;
if (!t) t = new node(i, i, v);
else if (t->l <= i && i <= t->r) t->upd(i, v);
else {
int L = l, R = r;
do {
if (i <= m) R = m;
else L = m + 1;
m = (L + R) >> 1;
} while ((i <= m) == (t->l <= m));
node *s = new node(L, R);
if (t->l <= m) s->lt = t;
else s->rt = t;
t = s;
s->upd(i, v);
}
x = gcd(lt ? lt->x: 0, rt ? rt->x: 0);
}
ll get(int lx, int rx) {
if (l > rx || r < lx) return 0;
if (l >= lx && r <= rx) return x;
return gcd(lt ? lt->get(lx, rx): 0, rt ? rt->get(lx, rx): 0);
}
};
struct segtree {
node root;
segtree *lt, *rt;
segtree() {
root = node(0, N - 1);
lt = rt = nullptr;
}
void upd(int l, int r, int i, int j, ll v) {
if (l != r) {
int m = (l + r) >> 1;
if (i <= m) {
if (!lt) lt = new segtree;
lt->upd(l, m, i, j, v);
} else {
if (!rt) rt = new segtree;
rt->upd(m + 1, r, i, j, v);
}
v = gcd(lt ? lt->root.get(j, j): 0, rt ? rt->root.get(j, j): 0);
}
root.upd(j, v);
}
ll get(int l, int r, int lx, int rx, int ly, int ry) {
if (l > rx || r < lx) return 0;
if (l >= lx && r <= rx) return root.get(ly, ry);
int m = (l + r) >> 1;
return gcd(lt ? lt->get(l, m, lx, rx, ly, ry): 0, rt ? rt->get(m + 1, r, lx, rx, ly, ry): 0);
}
};
struct segtree2d {
segtree root;
segtree2d() {
root = segtree();
}
void upd(int i, int j, ll v) {
root.upd(0, N - 1, i, j, v);
}
ll get(int lx, int rx, int ly, int ry) {
return root.get(0, N - 1, lx, rx, ly, ry);
}
} t;
void init(int n, int m) {}
void update(int i, int j, ll v) {
t.upd(i, j, v);
}
ll calculate(int lx, int ly, int rx, int ry) {
return t.get(lx, rx, ly, ry);
}
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