#include <bits/stdc++.h>
using namespace std;
#ifdef DEBUG
#include "../Library/debug.h"
#else
#include "Alice.h"
#define dbg(x...)
#endif
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef vector<int> vi;
typedef vector<bool> vb;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
#define FOR(i, a, b) for (int i = (a); i < (b); ++i)
#define F0R(i, a) for (int i = 0; i < (a); ++i)
#define FORd(i, a, b) for (inBobt i = (b) - 1; i >= (a); --i)
#define F0Rd(i, a) for (int i = (a) - 1; i >= 0; --i)
#define trav(a, x) for (auto& a : x)
#define f first
#define s second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) x.begin(), x.end()
const char nl = '\n';
const int INF = 1e9;
const ll MOD = 998244353;
const int mxN = 3e5+5;
#define DEBUG
#ifdef DEBUG
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
int N, M;
ll setN(int x){
N=x;
return M=uniform_int_distribution(1,int(1e18))(rng);
}
#endif
vpi Alice(){
int n = 5000;
ll x = setN(n);
vpi res;
FOR(i,2,n+1){
res.pb({(x%(i-1))+1,i});
}
return res;
}
namespace BOB{
ll ex_gcd(ll a,ll b,ll& x,ll& y){
if(b==0){
x=1;
b=0;
return a;
}
ll x1=0,y1=0;
ll d=ex_gcd(b,a%b,x1,y1);
x=y1;
y=x1-a/b*y1;
return d;
}
ll mul(ll a, ll b, ll m){
ll s=1;
if(b<0)s=-1,b*=-1;
ll ans=0;
while(b>0){
if(b%2==1)ans=(ans+s*a)%m;
b>>=1;
a=(a+a)%m;
}
return ans;
}
ll Bob(vpi v){
ll num[19];
num[0]=1;
F0R(i,18) num[i+1]=num[i]*10;
auto check=[&](ll a, ll b){
int x=0,y=0;
F0R(i,19)if(a<num[i]){
x=i;
break;
}
if(a&&!x) x=19;
if(b&&!y) y=19;
F0R(i,19)if(b<num[i]){
y=i;
break;
}
return (x+y)<=20;
};
ll res=0;
ll m=1;
trav(i,v){
i.f-=1;
i.s-=1;
ll x, y,d;
d=ex_gcd(m,i.s,x,y);
assert((res-i.f)%d==0);
if(!check(m,i.s))
return res;
// res = res+(i.f-res)/d*x%(i.s/d)*m;
res = res + mul(mul((i.f-res)/d, x,(i.s/d)),m,(m/d*i.s));
m = m/d*i.s;
res%=m;
if(res<0)res+=m;
}
return res;
}
}
#ifdef DEBUG
int main(){
auto gen=[&](ll a, ll b){
return uniform_int_distribution(a,b)(rng);
};
int cnt=0;
while(1){
vpi edges = Alice();
// dbg(edges)
shuffle(all(edges),rng);
int rem=gen(1,N/2);
dbg(++cnt,rem);
edges.erase(edges.end()-rem, edges.end());
// dbg(edges)
ll ans = BOB::Bob(edges);
if(ans != M){
dbg(N,M,ans);
return 0;
}
}
return 1;
}
#endif
#include <bits/stdc++.h>
using namespace std;
#ifdef DEBUG
#include "../Library/debug.h"
#else
#include "Bob.h"
#define dbg(x...)
#endif
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef vector<int> vi;
typedef vector<bool> vb;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
#define FOR(i, a, b) for (int i = (a); i < (b); ++i)
#define F0R(i, a) for (int i = 0; i < (a); ++i)
#define FORd(i, a, b) for (int i = (b) - 1; i >= (a); --i)
#define F0Rd(i, a) for (int i = (a) - 1; i >= 0; --i)
#define trav(a, x) for (auto& a : x)
#define f first
#define s second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) x.begin(), x.end()
const char nl = '\n';
const int INF = 1e9;
const int MOD = 1e9 + 7;
ll ex_gcd(ll a,ll b,ll& x,ll& y){
if(b==0){
x=1;
b=0;
return a;
}
ll x1=0,y1=0;
ll d=ex_gcd(b,a%b,x1,y1);
x=y1;
y=x1-a/b*y1;
return d;
}
ll mul(ll a, ll b, ll m){
ll s=1;
if(b<0)s=-1,b*=-1;
ll ans=0;
while(b>0){
if(b%2==1)ans=(ans+s*a)%m;
b>>=1;
a=(a+a)%m;
}
return ans;
}
ll Bob(vpi v){
ll num[19];
num[0]=1;
F0R(i,18) num[i+1]=num[i]*10;
auto check=[&](ll a, ll b){
int x=0,y=0;
F0R(i,19)if(a<num[i]){
x=i;
break;
}
if(a&&!x) x=19;
if(b&&!y) y=19;
F0R(i,19)if(b<num[i]){
y=i;
break;
}
return (x+y)<=20;
};
ll res=0;
ll m=1;
trav(i,v){
i.f-=1;
i.s-=1;
ll x, y,d;
d=ex_gcd(m,i.s,x,y);
assert((res-i.f)%d==0);
if(!check(m,i.s))
return res;
// res = res+(i.f-res)/d*x%(i.s/d)*m;
res = res + mul(mul((i.f-res)/d, x,(i.s/d)),m,(m/d*i.s));
m = m/d*i.s;
res%=m;
if(res<0)res+=m;
}
return res;
}
