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=gets.chomp.split(' ').map(&:to_i)
res = 0
buckets = Array.new(m, nil)
values = Array.new([m,n].min, 0)
buckets[x] = 0
values[0] = x
res += x
(1...n).each do |i|
  x = x * x % m
  values[i] = x
  if buckets[x].nil?
    buckets[x] = i
    res += x
  else
    j = buckets[x]
    d = i - j
    rotc = (n - i) / d
    rotv = values[j...i].sum
    res += rotc * rotv
    remain = n - (i + rotc * d)
    res += values[j...j+remain].sum
    break
  end
end
p res