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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::input;

fn main() {
  input!{
    n: usize,
    mut x: usize,
    m: usize,
  }
  
  let mut dp = vec![None;m];
  let mut val = x;
  let mut r = 0;
  let mut i = 0;
  while i < n {
    if let Some((c, v)) = dp[val] {
      let w = i - c;
      let l = n / w - 3;
      if l > 3 {
        r += (r-v)*l;
        i += w * l;
      }
      while i < n {
        r += val;
        i += 1;
        val = val * val % m;
      }
    } else {
      dp[val] = Some((i, r));
      r += val;
      i += 1;
      val = val * val % m;
    }
  }
  println!("{}", r);
}