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)
as = Array.new(M, nil)

c = X
m = 0
i = 0
g = false
while true do
  i += 1
  m += c
  if i == N then
    g = true
    puts m
    break
  end
  if as[c] == nil then
    as[c] = [i, m]
  else
    aa = as[c]
    ro = i - aa[0]
    rb = m - aa[1]
    loc = (N - i) / ro
    # pp [aa, i, m]
    i += loc * ro
    m += loc * rb
    break
  end
  c = c * c % M
end

if i == N then
  puts m
  g = true
end

unless g then
  while true do
    # puts i
    c = c * c % M
    i += 1
    m += c
    if i == N then
      puts m
      break
    end
  end
end