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: Accepted)

n, x, m = $stdin.readline.chomp.split.map(&:to_i)
dv = Array.new(m, 0)
used = Hash.new

res = 0
q = []
m.times do |i|
  if i == 0
    y = x
  else
    y = (dv[i - 1] ** 2) % m
  end

  if !used[y].nil?
    used[y].step(i - 1) do |j|
      q << dv[j]
    end
    n -= i
    break
  else
    dv[i] = y
    used[y] = i
    res += y
  end
end

q.size.times do |i|
  res += q[i] * ((n - i + q.size - 1) / q.size)
end
puts res