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 Perl (5.26.1) (judge verdict: Wrong Answer)

my ($N,$X,$M) =split(/ /,<>);

my $A = $X;
my $sum = $A%$M;

my %DB;
#print "1 $A\n";
for(my $i=1;$i<$N;$i++){
    my $mod = ($A*$A) % $M;
    #print "$i $mod\n";
    if(defined($DB{$mod}[0])){
      my $skip = $i - $DB{$mod}[0];
      my $B = int( ($N-$i-(($N-$i)%$skip))/$skip);
      $sum += ($sum-$DB{$mod}[1])*$B;
      $i += $skip * $B;
      
    }else{
      $DB{$mod}[0] = $i;
      $DB{$mod}[1] = $sum;
    }
    $sum += $mod;
    $A = $mod;
    
}
print "$sum";