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 = map(int, input().split())
Memo = [-1] * M
Cumsum = [0] * M
cumsum = X
x = X

for i in range(N):
    if Memo[x] != -1:
        break
    Memo[x] = i
    Cumsum[x] = cumsum
    x = x * x % M
    cumsum += x
else:
    print(cumsum-x)
    exit()
loop_length = i - Memo[x]
init_sum = Cumsum[x]
loop_sum = cumsum - init_sum

n_from_init = N - i - 1
n_loops, n_from_loopend = divmod(n_from_init, loop_length)
cumsum += n_loops * loop_sum

for i in range(n_from_loopend):
    x = x * x % M
    cumsum += x

print(cumsum)