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)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::ops::Bound::*;
use itertools::Itertools;
use num_traits::clamp;
use ordered_float::OrderedFloat;
use proconio::{input, marker::*, fastout};
use superslice::*;
//use ac_library_rs::*;

#[fastout]
fn main() {
    input! {
        n: u64, x: u64, m: u64,
    }

    let mut rems = HashMap::new();
    let mut sum = x;
    let mut a = x;
    rems.insert(a, (1, sum));
    let mut count = 1;
    let mut rep = None;
    while count < n {
        count += 1;
        a = a * a % m;
        sum += a;
        if let Some(&(idx, sum_low)) = rems.get(&a) {
            let sum_mid = sum - sum_low;
            let diff = count - idx;
            rep = Some((sum_low, sum_mid, diff, idx, a));
            break;
        } else {
            rems.insert(a, (count, sum));
        }
    }

    if rep.is_none() {
        println!("{}", sum);
        return;
    }

    let diff = rep.unwrap().2;
    let sum_low = rep.unwrap().0;
    let sum_mid = rep.unwrap().1;
    let low_idx = rep.unwrap().3;
    let num_mid = (n - low_idx) / diff ;
    let mut ans = sum_low + sum_mid * num_mid;
    let mut a = rep.unwrap().4;
    for _ in low_idx + num_mid * diff + 1..=n {
        a = a * a % m;
        ans += a;
    }
    println!("{}", ans);
}