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 seq = vec![0; m + 10];
    let mut flg = vec![0; m + 10];
    let (mut ls, mut le) = (1, 1);

    seq[1] = x;
    flg[x] = 1;
    for i in 2..=m+1 {
        seq[i] = (seq[i - 1] * seq[i - 1]) % m;
        if flg[seq[i]] > 0 {
            ls = flg[seq[i]];
            le = i;
            break;
        }
        flg[seq[i]] = i;
    }

    {
        // ~ ls
        let mut sum_uls = 0;
        let sum_ulsnum = ls - 1;
        for i in 1..ls {
            sum_uls += seq[i];
            if i == n {
                println!("{}", sum_uls);
                return;
            }
        }

        let mut sum_lse = 0;
        let sum_lsenum = le - ls;
        let mul_lse = (n - sum_ulsnum) / sum_lsenum;
        for i in ls..le {
            sum_lse += seq[i];
        }

        let remain = (n - sum_ulsnum) % sum_lsenum;
        let mut sum_remain = 0;
        for i in ls..ls + remain {
            sum_remain += seq[i];
        }

        println!("{}", sum_uls + sum_lse * mul_lse + sum_remain);
    }
}