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 solve() -> usize {
    input! {
        mut n: usize,
        mut x: usize,
        m: usize,
    }

    let mut used = vec![false; m];

    let mut ans = 0usize;
    while !used[x] {
        if n == 0 {
            return ans;
        }
        used[x] = true;
        ans += x;
        x = x * x % m;
        n -= 1;
    }

    let mut used = vec![false; m];
    let mut sum = 0;
    let mut le = 0;
    while !used[x] {
        if n == 0 {
            return ans + sum;
        }
        used[x] = true;
        le += 1;
        sum += x;
        x = x * x % m;
        n -= 1;
    }

    let t = n / le;
    n -= t * le;
    ans += sum * t + sum;

    while n > 0 {
        ans += x;
        x = x * x % m;
        n -= 1;
    }

    ans
}

fn main() {
    println!("{}", solve());
}