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: Accepted)

def main():
    n, x, m = map(int, input().split())
    a = x
    s = a
    seen = {a}
    seq = [a]
    for i in range(n - 1):
        if a == pow(a, 2, m):
            print(s + a * (n - i - 1))
            return
        else:
            a = pow(a, 2, m)
            if a in seen:
                si = seq.index(a)
                sl = i - si + 1
                ps = s - sum(seq[:si])
                ln = (n - i - 1) // sl
                lm = (n - i - 1) % sl
                print(s + ps * ln + sum(seq[si:si + lm]))
                return
            s += a
            seen.add(a)
            seq.append(a)
    print(s)


if __name__ == '__main__':
    main()