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)

N, X, M = [int(_) for _ in input().split()]

def f(x, m):
    return x % m

P = [-1 for _ in range(M+1)]

A = [X]
for i in range(N-1):
    v = f(A[-1] ** 2, M)
    A.append(v)
    if P[v] != -1:
        break
    P[v] = i

v = f(A[-1] ** 2, M)
idx = P[v]

S = sum(A[idx+1:])
q, r = divmod(N - len(A), len(A) - idx - 1)

ans = sum(A) + q * S + sum(A[idx+1:idx+1+r])
print(ans)