Problem:
Score : 500 points

Problem Statement

Let us denote by f(x, m) the remainder of the Euclidean division of x by m.

Let A be the sequence that is defined by the initial value A_1=X and the recurrence relation A_{n+1} = f(A_n^2, M).
Find \displaystyle{\sum_{i=1}^N A_i}.

Constraints

1 \leq N \leq 10^{10}

0 \leq X < M \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X M

Output

Print \displaystyle{\sum_{i=1}^N A_i}.

Sample Input 1

6 2 1001

Sample Output 1

1369

The sequence A begins 2,4,16,256,471,620,\ldots Therefore, the answer is 2+4+16+256+471+620=1369.

Sample Input 2

1000 2 16

Sample Output 2

6

The sequence A begins 2,4,0,0,\ldots Therefore, the answer is 6.

Sample Input 3

10000000000 10 99959

Sample Output 3

492443256176507

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

#define rep(i, n) for (ll i = 0; i < (n); i++)
using namespace std;
typedef long long ll;

int main() {
  ll n, x, m, cnt = 0, tot = 0, ans = 0, stot = 0;
  cin >> n >> x >> m;
  unordered_map<ll, ll> mp;
  vector<ll> v;
  while (mp.count(x) == 0) {
    mp[x] = cnt++;
    v.push_back(x);
    tot += x;
    x = x * x % m;
  }

  if (n <= cnt) {
    rep(i, n) ans += v[i];
    cout << ans << endl;
    return 0;
  }

  n -= mp[x];
  rep(i, v.size()) {
    if (v[i] == x) break;
    stot += v[i];
  }
  ll num = n / (cnt - mp[x]);
  ll mod = n % (cnt - mp[x]);
  ll pln = tot - stot;
  ans = pln * num + stot;
  for (int i = mp[x]; i < mp[x] + mod; ++i) ans += v[i];
  cout << ans << endl;
  return 0;
}