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, fastout};
use proconio::marker::{Chars};
use proconio::marker::Usize1;
use std::cmp::*;

macro_rules! debug {
      ($($a:expr),*) => {
          eprintln!(concat!($(stringify!($a), " = {:?}, "),*), $($a),*);
    }
}

#[fastout]
fn main() {
    input!{
        n: usize,
        x: usize,
        m: usize,
    }

    let mut ans = x;
    let mut sum = 0;
    let mut cnt = 0;
    let mut init = 0;
    let mut ok: bool = true;
    let mut dp: Vec<usize> = Vec::new();
    for i in 0..m {
        dp.push(ans);
        ans = (ans*ans)%m;
        for j in 0..dp.len() {
            if ans==dp[j] {
                cnt = dp.len() - j;
                init = j;
                ok = false;
                break;
            }
        }
        if ok==false {
            break;
        }
    }
    let mut init_sum = 0;
    for i in 0..init {
        init_sum += dp[i];
    }

    for i in init..dp.len() {
        sum += dp[i];
    }

    let d = (n-init) / cnt;
    let r = (n-init) % cnt;
    sum *= d;
    sum += init_sum;
    for i in 0..r {
        sum += ans;
        ans = (ans*ans)%m;
    }
    println!("{}", sum);
}