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: Runtime Error)

ans = []
ans[0] = a
a %= dev
for i in 1..(num - 1) do
  a = a ** 2
  a %= dev
  if ans.include?(a)
    unit = ans[ans.index(a) .. -1]
    unit_times = (num - i) / unit.size
    units_sum = unit.sum * unit_times
    rest_size = num - i - unit.size * unit_times
    rest = unit[0 .. (rest_size - 1)].sum
    ans[i] = units_sum + rest
    puts ans.sum
    return
  end
  ans[i] = a
end
puts ans.sum