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(clippy::needless_range_loop)]
#![allow(unused_macros)]
#![allow(dead_code)]
#![allow(unused_imports)]

use itertools::Itertools;
use proconio::input;
use proconio::marker::*;

use std::collections::BTreeMap;

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

    let mut map: BTreeMap<usize, usize> = BTreeMap::new();
    let mut cum = vec![0usize; m + 10];

    let mut i = 0;
    let mut ans = 0;
    let mut l = false;

    while i < n {
        if map.contains_key(&x) && !l {
            let j = map.get(&x).unwrap();
            let sum = cum[i] - cum[*j];
            let len = i - j;
            let cnt = (n - i) / len;
            n -= len * cnt;
            ans += sum * cnt;
            l = true;
            if n == i {
                break;
            }
        } else {
            map.insert(x, i);
            cum[i + 1] = cum[i] + x;
        }

        ans += x;
        x = x * x % m;
        i += 1;
    }

    println!("{}", ans);
}