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)

N, X, M = map(int, input().split())
loop_start = 0
n = 0
A = [X]
for i in range(1, N):
    tmp = A[-1]**2 % M
    if tmp not in A:
        A.append(tmp)
    else:
        loop_start = A.index(tmp)
        n = i - A.index(tmp)
        break
if n == 0:
    print(sum(A))
else:
    start, loop = A[:loop_start], A[loop_start:]
    sum_start = sum(start)
    sum_loop = sum(loop)
    div, mod = divmod(N - loop_start, n)
    end = loop[:mod]
    sum_end = sum(end)
    ans = sum_start + sum_loop * div + sum_end
    print(ans)