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())
dic = {}
dic[x]=0
ans = [0]*(10**5+1)
ans[0]=x
a = x

for i in range(1, n):
  a = (a**2) % m
  ans[i]=ans[i-1]+a
  if a in dic:
    kuri = i - dic[a]
    kurisum =ans[i]-ans[dic[a]]
    nokori = n-i
    time = nokori//kuri
    amari = nokori%kuri
    if dic[a] == 0:
      amarisum = ans[amari-1]
    else:
      amarisum = ans[dic[a]-1+amari]-ans[dic[a]-1]
    print(ans[i-1]+amarisum+time*kurisum)
    quit()
  else:
    dic[a]=i
print(ans[n-1])