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(unused_imports)]
use {
    itertools::Itertools,
    proconio::{fastout, input, marker::*},
    std::cmp::*,
    std::collections::*,
    std::io::Write,
    std::ops::*,
};

#[allow(unused_macros)]
macro_rules! dbg { ($($e:expr),*) => { #[cfg(debug_assertions)] $({ let (e, mut err) = (stringify!($e), std::io::stderr()); writeln!(err, "{} = {:?}", e, $e).unwrap() })* }; }

#[fastout]
fn main() {
    input! {
        mut n: usize,
        x: usize,
        m: usize,
    }
    let mut memo = vec![vec![(0, 0); m]; 35];
    memo[0] = (0..m).map(|i| (i, (i * i) % m)).collect();
    for d in 1..35 {
        for i in 0..m {
            let (c1, p) = memo[d - 1][i];
            let (c2, p) = memo[d - 1][p];
            memo[d][i] = (c1 + c2, p);
        }
    }
    let mut ans = 0;
    let mut x = x;
    for d in (0..35).rev() {
        if (n >> d) & 1 == 1 {
            ans += memo[d][x].0;
            x = memo[d][x].1;
        }
    }
    println!("{}", ans);
}