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 std::io::*;

fn main() {
    let mut s: String = String::new();
    std::io::stdin().read_to_string(&mut s).ok();
    let mut itr = s.trim().split_whitespace();
    let n: usize = itr.next().unwrap().parse().unwrap();
    let x: usize = itr.next().unwrap().parse().unwrap();
    let m: usize = itr.next().unwrap().parse().unwrap();

    let mut used: Vec<bool> = vec![false; m];
    let mut cycle: Vec<usize> = vec![];

    let mut start = 0;
    let mut now = x;
    for _ in 0..n {
        used[now] = true;
        cycle.push(now);
        now = now * now % m;
        if used[now] {
            for j in 0..cycle.len() {
                if cycle[j] == now {
                    start = j;
                    break;
                }
            }
            break;
        }
    }

    let mut s1 = 0;
    let mut s2 = 0;
    let len = cycle.len() - start;
    for i in 0..start {
        s1 += cycle[i];
    }
    for i in start..cycle.len() {
        s2 += cycle[i];
    }

    let mut ans = s1 + (n - start) / len * s2;
    for _ in 0..(n - start) % len {
        ans += now;
        now = now * now % m;
    }
    println!("{}", ans);
}