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())
a = [0 for i in range(M+1)]
b = [X]
a[X-1] = 1

for i in range(min(N-1, M)):
  X = X**2%M
  a[X-1] += 1
  b.append(X)
  if a[X-1] > 1:
    break

if len(b) == N:
  print(sum(b))

else:
  ls = b.index(b[-1])
  bloop = b[ls:-1]
  ans = 0
  if ls > 0:
    for i in range(ls):
      ans += b[i]
  loop = (N-ls)//len(bloop)
  ans += sum(bloop) * loop
  if (N-ls)%len(bloop) != 0:
    for i in range((N-ls)%len(bloop)):
      ans += bloop[i]
  print(ans)