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 Ruby (2.7.1) (judge verdict: Wrong Answer)

n, x, m = gets.split.map(&:to_i)

r = x
res = x
r_arr = [r]
r_hash = {r => 0}

(n-1).times do |i|
  r = r ** 2 % m
  break unless r_hash[r].nil?

  res += r
  r_arr << r
  r_hash[r] = i + 1
end

ind = r_hash[r]
sub_arr = r_arr[ind..-1]
sum = sub_arr.sum
count = sub_arr.length
res += ((n - r_arr.length) / count) * sum

de = (n - r_arr.length) % count
res += r
(de - 1).times do 
  r = r ** 2 % m
  res += r
end
puts res