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)

fn main() {
    proconio::input! {
        n: u64,
        mut x: u64,
        m: u64,
    }

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