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)

n, x, m = map(int, input().split())
nn = n

slow = x
fast = x
out = 0

while n:
    if n != nn and slow == fast:
        break
    out += slow
    n -= 1

    slow = (slow * slow) % m
    fast = (fast * fast) % m
    fast = (fast * fast) % m

head = slow
curr_sum = slow
curr_count = 1
slow = (slow * slow) % m
while slow != head:
    curr_count += 1
    curr_sum += slow
    slow = (slow * slow) % m

while n % curr_count:
    out += slow
    n -= 1
    slow = (slow * slow) % m

out += (n // curr_count) * curr_sum
print(out)