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 Python (3.8.2) (judge verdict: Accepted)

n, x, mod = [int(i) for i in input().split()]
ans = x
tmp = x
s = set()
s.add(x)
for i in range(1, n):
    tmp = (tmp * tmp) % mod
    if tmp in s:
        cycle = [tmp]
        t = (tmp * tmp) % mod
        while t != tmp:
            cycle.append(t)
            t = (t * t) % mod
        ans += sum(cycle) * ((n - i) // len(cycle))
        ans += sum(cycle[:(n-i)%len(cycle)])
        break
    else:
        s.add(tmp)
        ans += tmp
print(ans)