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.split.map(&:to_i)

visited = [nil] * M
chain = []

seconds = 0
while visited[x].nil?
    chain << x
    visited[x] = seconds
    x = x * x % M
    seconds += 1
end

start = visited[x]
length = seconds - start

chain_sum = [0]
for e in chain
    chain_sum << chain_sum[-1] + e
end

answer = 0
if n < start
    answer += chain_sum[n]
    n = 0
else
    answer += chain_sum[start]
    n -= start
end

answer += (chain_sum[seconds] - chain_sum[start]) * (n / length)
answer += chain_sum[start + (n % length)] - chain_sum[start]
puts answer