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)

#![allow(unused_imports)]

use proconio::{input, fastout};
use proconio::marker::*;

#[fastout]
fn main() {
    input! {
    n: usize,
    x: usize,
    m: usize
    }
    let mut dm = vec![0; m];
    let mut s = vec![0; m + 2];
    for i in 0..m {
        dm[i] = i * i % m;
    }
    let (mut l, mut r) = (0, 0);
    let mut nums = vec![0; m + 1];
    let mut t = x;
    for i in 1..=m + 1 {
        s[i] = s[i - 1] + t;
        if nums[t] > 0 {
            l = nums[t] - 1;
            r = i - 1;
            break;
        }
        nums[t] = i;
        t = dm[t];
    }
    if n <= r {
        print!("{}", s[n]);
        return;
    }
    let mut ans: usize = s[l];
    ans += (n - l) / (r - l) * (s[r] - s[l]) + s[l + (n - l) % (r - l)] - s[l];
    println!("{}", ans);
}