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())
if M==1:
  print(0)
  exit()
lis = [0]*M
lis[0] = X
check = [-1]*M
check[X] = 0
for i in range(1,M):
  lis[i] = pow(lis[i-1],2,M)
  if check[lis[i]]>=0:
    cyc = i-check[lis[i]]
    start = check[lis[i]]
    break
  check[lis[i]] = i
from itertools import groupby, accumulate, product, permutations, combinations
cum = list(accumulate(lis))
cyc_cum = cum[-1]-cum[start]
if N-1<=start+cyc:
  ans = cum[N-1]
else:
  ans = cyc_cum*((N-1-start)//cyc)+cum[(N-1-start)%cyc+start]
print(ans)