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 input_mcr::*;
use std::cmp::*;

pub trait CycleSolver {
    fn next(&self, i: usize) -> usize;
    fn get_head_cycle_and_tail(
        &mut self,
        head: &[usize],
        cycle: &[usize],
        cycle_times: usize,
        tail: &[usize],
    );
    fn solve(&mut self, n: usize, ini: usize, len: usize) {
        assert!(ini < n);
        if len == 0 {
            return;
        }
        let mut visited = vec![false; n];
        visited[ini] = true;
        let mut v = ini;
        let mut i = 1;
        let mut history = vec![];
        history.reserve(n);
        history.push(ini);
        while i < len {
            v = self.next(v);
            assert!(v < n);
            if visited[v] {
                let cycle_begin = history.iter().position(|&x| x == v).unwrap();
                let cycle_len = i - cycle_begin;
                let rem = len - i - 1;
                let cycle_times = rem / cycle_len;
                i += cycle_times * cycle_len;
                let rem = len - i;
                self.get_head_cycle_and_tail(
                    &history[..cycle_begin],
                    &history[cycle_begin..],
                    cycle_times + 1,
                    &history[cycle_begin..cycle_begin + rem],
                );
                return;
            } else {
                history.push(v);
                visited[v] = true;
                i += 1;
            }
        }
        self.get_head_cycle_and_tail(&history, &[], 0, &[]);
    }
}

struct Solver {
    res: usize,
    m: usize,
}

impl CycleSolver for Solver {
    #[inline]
    fn next(&self, i: usize) -> usize {
        i * i % self.m
    }
    fn get_head_cycle_and_tail(
        &mut self,
        head: &[usize],
        cycle: &[usize],
        cycle_times: usize,
        tail: &[usize],
    ) {
        // eprintln!("head = {:?}, cycle = {:?}", head, cycle);
        let s = head.iter().sum::<usize>();
        let t = cycle.iter().sum::<usize>();
        let u = tail.iter().sum::<usize>();
        self.res += s + t * cycle_times + u;
    }
}

fn main() {
    input! {
        n: usize,
        x: usize,
        m: usize,
    }
    let mut solver = Solver { res: 0, m };
    solver.solve(m, ((x % m) * x) % m, n - 1);
    let res = x + solver.res;
    println!("{}", res);
}
pub mod input_mcr {
    // ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
    #[macro_export(local_inner_macros)]
    macro_rules! input {
        (source = $s:expr, $($r:tt)*) => {
            let mut parser = Parser::from_str($s);
            input_inner!{parser, $($r)*}
        };
        (parser = $parser:ident, $($r:tt)*) => {
            input_inner!{$parser, $($r)*}
        };
        (new_stdin_parser = $parser:ident, $($r:tt)*) => {
            let stdin = std::io::stdin();
            let reader = std::io::BufReader::new(stdin.lock());
            let mut $parser = Parser::new(reader);
            input_inner!{$parser, $($r)*}
        };
        ($($r:tt)*) => {
            input!{new_stdin_parser = parser, $($r)*}
        };
    }

    #[macro_export(local_inner_macros)]
    macro_rules! input_inner {
        ($parser:ident) => {};
        ($parser:ident, ) => {};
        ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
            let $var = read_value!($parser, $t);
            input_inner!{$parser $($r)*}
        };
    }

    #[macro_export(local_inner_macros)]
    macro_rules! read_value {
        ($parser:ident, ( $($t:tt),* )) => {
            ( $(read_value!($parser, $t)),* )
        };
        ($parser:ident, [ $t:tt ; $len:expr ]) => {
            (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
        };
        ($parser:ident, chars) => {
            read_value!($parser, String).chars().collect::<Vec<char>>()
        };
        ($parser:ident, char_) => {
            read_value!($parser, String).chars().collect::<Vec<char>>()[0]
        };
        ($parser:ident, usize1) => {
            read_value!($parser, usize) - 1
        };
        ($parser:ident, line) => {
            $parser.next_line()
        };
        ($parser:ident, line_) => {
            $parser.next_line().chars().collect::<Vec<char>>()
        };
        ($parser:ident, $t:ty) => {
            $parser.next::<$t>().expect("Parse error")
        };
    }

    use std::collections::VecDeque;
    use std::io;
    use std::io::BufRead;
    use std::str;

    pub struct Parser<R> {
        pub reader: R,
        buf: VecDeque<u8>,
        parse_buf: Vec<u8>,
    }

    impl Parser<io::Empty> {
        pub fn from_str(s: &str) -> Parser<io::Empty> {
            Parser {
                reader: io::empty(),
                buf: VecDeque::from(s.as_bytes().to_vec()),
                parse_buf: vec![],
            }
        }
    }

    impl<R: BufRead> Parser<R> {
        pub fn new(reader: R) -> Parser<R> {
            Parser {
                reader: reader,
                buf: VecDeque::new(),
                parse_buf: vec![],
            }
        }
        pub fn update_buf(&mut self) {
            loop {
                let (len, complete) = {
                    let buf2 = self.reader.fill_buf().unwrap();
                    self.buf.extend(buf2.iter());
                    let len = buf2.len();
                    (len, buf2.last() < Some(&0x20))
                };
                self.reader.consume(len);
                if complete {
                    break;
                }
            }
        }
        pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
            loop {
                while let Some(c) = self.buf.pop_front() {
                    if c > 0x20 {
                        self.buf.push_front(c);
                        break;
                    }
                }
                self.parse_buf.clear();
                while let Some(c) = self.buf.pop_front() {
                    if c <= 0x20 {
                        self.buf.push_front(c);
                        break;
                    } else {
                        self.parse_buf.push(c);
                    }
                }
                if self.parse_buf.is_empty() {
                    self.update_buf();
                } else {
                    return unsafe { str::from_utf8_unchecked(&self.parse_buf) }.parse::<T>();
                }
            }
        }
        pub fn next_line(&mut self) -> String {
            loop {
                while let Some(c) = self.buf.pop_front() {
                    if c >= 0x20 {
                        self.buf.push_front(c);
                        break;
                    }
                }
                self.parse_buf.clear();
                while let Some(c) = self.buf.pop_front() {
                    if c < 0x20 {
                        self.buf.push_front(c);
                        break;
                    } else {
                        self.parse_buf.push(c);
                    }
                }
                if self.parse_buf.is_empty() {
                    self.update_buf();
                } else {
                    return unsafe { str::from_utf8_unchecked(&self.parse_buf) }.to_string();
                }
            }
        }
    }
}