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 PHP (7.4.4) (judge verdict: Accepted)

<?php


[$N, $X, $M] = fscanf(STDIN, "%d%d%d");

$list = [];
$sum = array_fill(0,100000,0);
$N--;
$ans = $X;
$sum[0] = $X;
$list[$X] = 0;
$now = 1;
$n = $X;
while($N--){
    $n = ($n * $n) % $M;
    if(isset($list[$n])){
        $zan = $N + 1;
        $per = $zan % ($now - $list[$n]);
        $mul = intdiv($zan,  ($now - $list[$n]));
        $ans += $mul * ($sum[$now-1] - $sum[$list[$n] -1]);
        $ans += $sum[$list[$n] -1 + $per] - $sum[$list[$n] -1];
        echo $ans;
        exit;
    }else{
        if($n === 0){
            echo $ans;
            exit;
        }
        $list[$n] = $now;
        $ans += $n;

        $sum[$now] = $ans;
        $now++;
    }
}
echo $ans;