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())

exist = [0 for _ in range(M)]


def f(a, m):
  return a % m

S = 0
A = X
ls = []
for n in range(1, N + 1):
  if exist[A] == 0:
    exist[A] = n
    ls.append(A)
  else:
    term = n - exist[A]
    loop = ls[-term:]
    res = N - n + 1
    S += (res // term) * sum(loop) + sum(loop[:res % term])
    break
  S += A
  A = f(A**2, M)

print(S)