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)

def main():
    n,x,m=map(int,input().split())
    A=[]
    s=set()
    while True:
        A.append(x)
        s.add(x)
        x**=2
        x%=m
        if x==0:
            A.append(0)
            break
        if x in s:
            break
    #print(A)
    if n<=len(A):
        return sum(A[:n])
    if A[-1]==0:
        return sum(A)
    for idx,a in enumerate(A):
        if a==x:
            break
    l=len(A)-idx
    cycles=(n-idx)//l
    remain=(n-idx)%l
    #print(idx,l,cycles)
    s=sum(A[idx:])
    ans=sum(A[:idx])
    ans+=s*cycles
    ans+=sum(A[idx:idx+remain])
    return ans
    
if __name__=='__main__':
    print(main())