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)

from collections import defaultdict
from collections import deque
from collections import Counter
import math
import itertools

def readInt():
  return int(input())
def readInts():
  return list(map(int, input().split()))
def readChar():
  return input()
def readChars():
  return input().split()

n,x,m = readInts()
an = x
ans = an
moddict = {an:1}
for i in range(2,n+1):
  an = pow(an,2,m)
  if an in moddict:
    break
  else:
    moddict[an]=i
  ans += an
#print(ans)
if an in moddict:
  rep = (n-len(moddict))//(len(moddict)-moddict[an]+1)
  rem = (n-len(moddict))%(len(moddict)-moddict[an]+1)
  #print(rep,rem)
  rep_sum = 0
  boo = 0
  for key in moddict:
    if boo or key==an:
      boo = 1
      rep_sum+=key

  ans += rep_sum*rep
  for i in range(rem):
    ans+=an
    an=pow(an,2,m)


print(ans)