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.Linq;
using System.ComponentModel;
using System.Collections.Generic;

namespace AtCoder {
    public class TestCode {
        static void Main(string[] args) {
            var (N, X, M) = ReadValue<long, long, long>();
            var tmp = new long[M];
            tmp[X] = X;
            var t = new List<long>();
            long ans = X;
            long x = X;
            for(var i = 1; i < N; i++) {
                x = (x * x) % M;
                if (tmp[x] == -1) {
                    var a = (N - i) / t.Count();
                    ans += a * t.Sum() + t.Take((int)((N - i) % t.Count())).Sum();
                    break;
                } else if (tmp[x] == 0) {
                    if (x == 0) break;
                    tmp[x] = x;
                    ans += x;
                } else {
                    tmp[x] = -1;
                    ans += x;
                    t.Add(x);
                }
            }
            Console.Write(ans);
        }

        static T1 ReadValue<T1>() {
            var t1 = TypeDescriptor.GetConverter(typeof(T1));
            return (T1)t1.ConvertFromString(Console.ReadLine());
        }

        static (T1, T2) ReadValue<T1, T2>() {
            var rl = Console.ReadLine().Trim().Split().ToArray();
            var t1 = TypeDescriptor.GetConverter(typeof(T1));
            var t2 = TypeDescriptor.GetConverter(typeof(T2));
            return ((T1)t1.ConvertFromString(rl[0]),
                    (T2)t2.ConvertFromString(rl[1]));
        }

        static (T1, T2, T3) ReadValue<T1, T2, T3>() {
            var rl = Console.ReadLine().Trim().Split().ToArray();
            var t1 = TypeDescriptor.GetConverter(typeof(T1));
            var t2 = TypeDescriptor.GetConverter(typeof(T2));
            var t3 = TypeDescriptor.GetConverter(typeof(T3));
            return ((T1)t1.ConvertFromString(rl[0]),
                    (T2)t2.ConvertFromString(rl[1]),
                    (T3)t3.ConvertFromString(rl[2]));
        }

        static (T1, T2, T3, T4) ReadValue<T1, T2, T3, T4>() {
            var rl = Console.ReadLine().Trim().Split().ToArray();
            var t1 = TypeDescriptor.GetConverter(typeof(T1));
            var t2 = TypeDescriptor.GetConverter(typeof(T2));
            var t3 = TypeDescriptor.GetConverter(typeof(T3));
            var t4 = TypeDescriptor.GetConverter(typeof(T4));
            return ((T1)t1.ConvertFromString(rl[0]),
                    (T2)t2.ConvertFromString(rl[1]),
                    (T3)t3.ConvertFromString(rl[2]),
                    (T4)t4.ConvertFromString(rl[3]));
        }

        static (T1, T2, T3, T4, T5) ReadValue<T1, T2, T3, T4, T5>() {
            var rl = Console.ReadLine().Trim().Split().ToArray();
            var t1 = TypeDescriptor.GetConverter(typeof(T1));
            var t2 = TypeDescriptor.GetConverter(typeof(T2));
            var t3 = TypeDescriptor.GetConverter(typeof(T3));
            var t4 = TypeDescriptor.GetConverter(typeof(T4));
            var t5 = TypeDescriptor.GetConverter(typeof(T5));
            return ((T1)t1.ConvertFromString(rl[0]),
                    (T2)t2.ConvertFromString(rl[1]),
                    (T3)t3.ConvertFromString(rl[2]),
                    (T4)t4.ConvertFromString(rl[3]),
                    (T5)t5.ConvertFromString(rl[4]));
        }

        static string[] stringArray => Console.ReadLine().Split().ToArray();
        static int[] intArray => Console.ReadLine().Trim().Split().Select(int.Parse).ToArray();
        static long[] longArray => Console.ReadLine().Trim().Split().Select(long.Parse).ToArray();
        static double[] doubleArray => Console.ReadLine().Trim().Split().Select(double.Parse).ToArray();
        static decimal[] decimalArray => Console.ReadLine().Trim().Split().Select(decimal.Parse).ToArray();
    }
}