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 PyPy3 (7.3.0) (judge verdict: Accepted)

import math
from functools import reduce
from collections import deque
import sys
sys.setrecursionlimit(10**7)

def input():
    return sys.stdin.readline().strip()

# スペース区切りの入力を読み込んで数値リストにして返します。
def get_nums_l():
    return [ int(s) for s in input().split(" ")]

# 改行またはスペース区切りの入力をすべて読み込んでイテレータを返します。
def get_all_int():
    return map(int, open(0).read().split())

def log(*args):
    print("DEBUG:", *args, file=sys.stderr)

def main():

    n,x,MOD = get_nums_l()

    ans = x
    now = x

    known = {x}
    history = [x]

    for i in range(1, n):
        y = now**2 % MOD
        if y in known:
            # ループ長と合計を検出する
            last_known = i-1
            loop_sum = y
            while history[last_known] != y:
                loop_sum += history[last_known]
                last_known -= 1
            loop_len = i - last_known

            ans += (n-i) // loop_len * loop_sum

            # log(loop_len, loop_sum)

            for j in range((n-i) % loop_len):
                if (last_known + j) < len(history):
                    ans += history[last_known + j]
            break

        known.add(y)
        history.append(y)
        ans += y
        now = y
    
    print(ans)


main()