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,
        x: usize,
        m: usize,
    }
    
    let mut to = vec![100001; m];
    let l = {
        let mut a = x;
        while to[a] == 100001 {
            let b = a * a % m;
            to[a] = b;
            a = b;
        }
        a
    };
    
    let mut ans = 0;
    let mut a = x;
    if n <= 300000 {
        for _ in 0..n {
            ans += a;
            a = a * a % m;
        }
    }
    else {
        let mut n = n;
        while a != l {
            ans += a;
            a = a * a % m;
            n -= 1;
        }
        // a = a * a % m;
        // n -= 1;
        let mut cycle = 0;
        let mut sum = 0;
        loop {
            sum += a;
            a = a * a % m;
            cycle += 1;
            if a == l {
                break;
            }
        }
        let loops = n/cycle;
        ans += sum * loops;
        n -= loops * cycle;
        for _ in 0..n {
            ans += a;
            a = a * a % m;
        }
    }
    println!("{}", ans);
}