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)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use competitive::prelude::*;

// #[argio(output = AtCoder)]
// fn main(n: usize, mut x: usize, m: usize) -> usize {
//     let mut prev = vec![None; m];
//     let mut acc = 0;

//     for i in 0..n {
//         if let &Some((prev_i, prev_acc)) = &prev[x] {
//             let rest = n - i;
//             let loop_len = i - prev_i;
//             let loop_sum = acc - prev_acc;

//             let loop_cnt = rest / loop_len;
//             acc += loop_sum * loop_cnt;
//             let rest = rest - loop_cnt * loop_len;

//             for _ in 0..rest {
//                 acc += x;
//                 x = x * x % m;
//             }

//             break;
//         } else {
//             prev[x] = Some((i, acc));
//             acc += x;
//             x = x * x % m;
//         }
//     }

//     acc
// }

#[argio(output = AtCoder)]
fn main(n: usize, mut x: usize, m: usize) -> usize {
    let mut x1 = x;
    let mut acc1 = x;

    let mut x2 = x * x % m;
    let mut acc2 = x1 + x2;

    for i in 1..n {
        if x1 == x2 {
            let loop_len = i;
            let loop_sum = acc2 - acc1;
            let rest = n - i;
            let loop_cnt = rest / loop_len;
            acc1 += loop_sum * loop_cnt;
            let rest = rest - loop_cnt * loop_len;

            for _ in 0..rest {
                x1 = x1 * x1 % m;
                acc1 += x1;
            }

            break;
        }

        x1 = x1 * x1 % m;
        acc1 += x1;

        x2 = x2 * x2 % m;
        acc2 += x2;
        x2 = x2 * x2 % m;
        acc2 += x2;
    }
    acc1

    // let mut prev = vec![None; m];
    // let mut acc = 0;

    // for i in 0..n {
    //     if let &Some((prev_i, prev_acc)) = &prev[x] {
    //         let rest = n - i;
    //         let loop_len = i - prev_i;
    //         let loop_sum = acc - prev_acc;

    //         let loop_cnt = rest / loop_len;
    //         acc += loop_sum * loop_cnt;
    //         let rest = rest - loop_cnt * loop_len;

    //         for _ in 0..rest {
    //             acc += x;
    //             x = x * x % m;
    //         }

    //         break;
    //     } else {
    //         prev[x] = Some((i, acc));
    //         acc += x;
    //         x = x * x % m;
    //     }
    // }

    // acc
}
*/

fn main() {
    let exe = "/tmp/bin69EE1A17";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";