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)

# frozen_string_literal: true

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

a = x

h = Hash.new { false }
l = -1
b = []
n.times do
  if h[a]
    l = a
    break
  end
  h[a] = true
  b.push a
  a = (a * a) % m
end

b1, b2 = b.slice_before(l).to_a
b1 = [] if b1.nil?
b2 = [] if b2.nil?
remain = n - b.size
if l == b[0]
  b2 = b1
  b1 = []
end
d = 0
dd = 0
if remain.positive?
  unless b2.empty?
    d = remain / b2.size
    dd = remain % b2.size
  end
end

sum1 = b1.sum
sum2 = b2.sum * (d + 1)
sum = sum1 + sum2 + b2.take(dd).sum
puts sum