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 PyPy3 (7.3.0) (judge verdict: Accepted)

n, x, m = map(int, input().split())
ans = 0
if x == 0:
    print(0)
    exit()
elif x == 1:
    print(n)
    exit()

flag = [False]*m
a = []
while not flag[x]:
    flag[x] = True
    a.append(x)
    x = pow(x, 2, m)
loop_start_idx = a.index(x)
loop_len = len(a) - loop_start_idx
loop_count = (n - loop_start_idx)//loop_len
loop_amari = (n-loop_start_idx)%loop_len
ans = sum(a[:loop_start_idx])
ans += sum(a[loop_start_idx:])*loop_count
ans += sum(a[loop_start_idx: loop_start_idx + loop_amari])
print(ans)