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())
A = X
T = [A]
ans = A

for n in range(1, N):
  tmp = A ** 2 % M
    
  if tmp == 0:
    break
    
  if tmp in T:    
    pre_tmp_idx = T.index(tmp)
    syuki = T[pre_tmp_idx:]
    
    len_syuki = len(syuki)
    
    nokori_syuki = (N - n) // len_syuki    
    syuki_amari = (N - n) % len_syuki
    
    ans += sum(syuki) * nokori_syuki + sum(syuki[:syuki_amari])
    break

  ans += tmp
  A = tmp
  T.append(tmp)
    
print(ans)