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;
use std::collections::HashMap;

fn main() {
    input! {
        n: usize,
        x: usize,
        m: usize,
    }

    let mut hm = HashMap::new();
    let mut ss = vec![0];
    let mut a = x;
    let mut s = 0;

    for i in 1..=n {
        if let Some(j) = hm.get(&a) {
            let loop_size = i - j;
            let s_offset = ss.get(j - 1).unwrap();
            let mut ans = (s - s_offset) * ((n - (j - 1)) / loop_size);
            ans += s_offset;
            let r = (n - (j - 1)) % loop_size;
            ans += ss.get(j - 1 + r).unwrap() - s_offset;
            println!("{}", ans);
            return;
        }
        s += a;
        hm.insert(a, i);
        ss.push(s);
        a = (a * a) % m;
    }
    println!("{}", s);
}