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! {
        nxm: (u64, usize, usize),
    };
    let n = nxm.0;
    let x = nxm.1;
    let m = nxm.2;
    let mut v: Vec<usize> = vec![0;m];
    let mut p: Vec<u64> = vec![0;m];
    let mut a = x;
    v[a] = 1;
    p[a] = a as u64;
    let mut sum_all = a as u64;
    let mut i = 1;
    let mut s;
    let mut t;
    let mut sum_start: u64;
    let mut sum_loop: u64;
    loop {
        if i as u64 >= n {
            println!("{}", sum_all);
            return;
        }
        i += 1;
        a = (a * a) % m;
        sum_all += a as u64;
        if v[a] == 0 {
            v[a] = i;
            p[a] = sum_all;
        } else {
            s = v[a];
            t = i;
            sum_start = p[a];
            sum_loop = sum_all - sum_start;
            break;
        }
    }
    let cnt_last = (n - s as u64) % (t as u64 - s as u64);
    let cnt_loop = (n - s as u64) / (t as u64 - s as u64);
    let sum_loops = sum_loop * cnt_loop;
    let mut sum_last = 0;
    for _ in 0..cnt_last {
        a = (a * a) % m;
        sum_last += a as u64;
    }
    println!("{}", sum_start + sum_loops + sum_last);
}