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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import sys

N, X, M = map(int, input().split())

A = [X]
for i in range(1, N):
    X = (X ** 2) % M
    # print(X)
    if X in A:
        pattern_start_id = A.index(X)
        break
    else:
        A.append(X)
else:
    ans = sum(A)
    print(ans)
    sys.exit()

before_pattern = A[:pattern_start_id]
pattern = A[pattern_start_id:]

ans = sum(before_pattern) + sum(pattern) * ((N - len(before_pattern)) // len(pattern)) + sum(pattern[:(N - len(before_pattern)) % len(pattern)])

print(ans)