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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;

namespace ConsoleApp1
{
    class ABC179E
    {
        // https://atcoder.jp/contests/abc179/tasks/abc179_e

        static void Main(string[] args)
        {
            var items = Console.ReadLine().Split();
            long N = Int64.Parse(items[0]);
            long X = Int64.Parse(items[1]);
            long M = Int64.Parse(items[2]);

            long sum = X;
            long Ai = X;

            if (X < 1)
            {
                Console.WriteLine(0);
                return;
            }

            long[] mo = new long[M];
            long[] Si = new long[M];
            mo[Ai] = 1;
            Si[1] = Ai;

            for (int i = 2; i <= N; i++)
            {
                Ai = Ai * Ai % M;

                if (mo[Ai] > 0)
                {
                    long loopSize = i - mo[Ai];

                    if ((N - i + 1) % loopSize == 0)
                    {
                        long loopCount = (N - i + 1) / loopSize;
                        long loopSum = Si[i - 1] - Si[mo[Ai] - 1];
                        sum += loopSum * loopCount;

                        Console.WriteLine(sum);
                        return;
                    }
                }

                mo[Ai] = i;
                sum += Ai;
                Si[i] = sum;
            }

            Console.WriteLine(sum);
        }
    }
}