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, m = map( int, input().split() )
# n, x, m = 10000000000, 10, 99959

found = [ -1 for _ in range( m ) ]
summation = [ 0 ]
a = x
length, loop_s, s_0, start = -1, -1, -1, -1
for i in range( n ):
    if found[ a ] > -1:
        length = i - found[ a ]
        loop_s = summation[ i ] - summation[ found[ a ] ]
        start = found[ a ]
        s_0 = summation[ start ]
        # print( length, loop_s, s_0, a, start )
        break
    else:
        found[ a ] = i
    # print( i, a )
    summation.append( summation[ -1 ] + a )
    a = ( a * a ) % m

if length < 0:
    print( summation[ n ] )
else:
    times = ( n - start ) // length
    rem = ( n - start ) % length
    rem_s = summation[ start + rem ] - s_0
    print( s_0 + loop_s * times + rem_s )