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.IO;
using System.Linq;
using System.Collections.Generic;
using static System.Math;
using static System.Console;
using static System.Linq.Enumerable;
using static System.Numerics.BitOperations;

namespace ABC179{
    class SolverE{
        static void Main(){
            SetOut(new StreamWriter(Console.OpenStandardOutput()){AutoFlush = false});
            new SolverE().Solve();
            Out.Flush();
        }

        public void Solve(){
            checked{
                (long N, int X, int M) = Get.Tuple<long,int,int>();
                List<int> a = new List<int>();
                int x = X;
                int[] id = Enumerable.Repeat<int>(-1,M).ToArray();
                int len = 0;
                while(id[x] == -1){
                    id[x] = len;
                    a.Add(x);
                    x = (int)((long)x * (long)x % M);
                    len++;
                }
                int istart = id[x];
                var ilast = len - 1;
                int llen = ilast - (istart - 1);
                long lsum = 0;
                // Debug.Put(istart,"istart",ilast,"ilast",a.ToArray(),"a");
                for(int i=istart; i<=ilast; i++){
                    lsum += a[i];
                }

                if(N <= ilast){
                    long t = 0;
                    for(int i=0; i<N; i++){
                        t += a[i];
                    }
                    WriteLine(t);
                    return;
                }

                long tot = 0;
                for(int i=0; i<istart; i++){
                    tot += a[i];
                }
                tot += lsum * ((N - istart) / (llen));
                int rlen = (int)((N - istart) % llen);
                for(int i=0; i<rlen; i++){
                    tot += a[i+istart];
                }
                WriteLine(tot);
            }
        }



        static class Mod{
            public static long Pow(long x, long e, long mod = long.MaxValue){
                long res = 1;
                while (e > 0){
                    if ((e & 1) == 1) res = res * x % mod;
                    x = x * x % mod;
                    e >>= 1;
                }
                return res;
            }

            // 逆元を求める。前提：modが素数、aがpの倍数でない。フェルマーの小定理に基づく。
            public static long Inv(long a, long mod){
                return Pow(a, mod-2, mod);
            }
        }
        
        static class Debug{
            public static void Put(object obj, int padLeft = 0, bool newline = true){

                if (obj is Array arr){
                    if( arr.Rank == 1 ){
                        Write("[");
                        for(int i=0; i<arr.Length; i++){
                            Put(arr.GetValue(i), padLeft, false);
                            if( i < arr.Length - 1 ) Write(", ");
                        }
                        Write("]");
                    }
                    else if( arr.Rank == 2 ){
                        Write("[\n");
                        for(int i=0; i<arr.GetLength(0); i++){
                            Write(" [");
                            for(int j=0; j<arr.GetLength(1); j++){
                                Put(arr.GetValue(i,j), padLeft, false);
                                if (j < arr.GetLength(1) - 1) Write(", ");
                            }
                            if (i < arr.GetLength(0) - 1 ) Write("],\n");
                            else Write("]\n");
                        }
                        Write("]");
                    }
                }
                else if( obj is ValueType val ){
                    Write(val.ToString().PadLeft(padLeft));
                }
                else if( obj is string str ){
                    Write(str.PadLeft(padLeft));
                }
                else{
                    Write(obj.ToString().PadLeft(padLeft));
                }

                if(newline) Write("\n");
            }

            public static void Put(object obj, string label, int padLeft = 0, bool newline = true){
                Write("\u001b[32m{0}:\u001b[0m ", label);
                Put(obj, padLeft, newline);
            }
            public static void Put(params object[] args){
                if (args.Length % 2 != 0){
                    WriteLine("Debug.Put(params): arg length shall be multiple of 2.");
                }
                for(int i=0; i<args.Length; i += 2){
                    Put(args[i], (string)args[i+1],0, false);
                    Write("   ");
                }
                Write("\n");
            }
        }

        
        private static class Get{
            public static string Str() => ReadLine().Trim();
            public static int Int() => int.Parse(Str());
            public static long Long() => long.Parse(Str());
            public static double Double() => double.Parse(Str());
            public static string[] Strs() => Str().Split(' ', StringSplitOptions.RemoveEmptyEntries);
            public static int[] Ints() => Strs().Select(int.Parse).ToArray();
            public static long[] Longs() => Strs().Select(long.Parse).ToArray();
            public static double[] Doubles() => Strs().Select(double.Parse).ToArray();
            static bool TypeEq<T,U>() => typeof(T).Equals(typeof(U));
            static T TypeConv<T,U>(U u) => (T)Convert.ChangeType(u, typeof(T));
            static T TypeConv<T>(string s) => TypeEq<T, int>() ?   TypeConv<T, int>(int.Parse(s))
                                        : TypeEq<T, long>() ?       TypeConv<T, long>(long.Parse(s))
                                        : TypeEq<T, double>() ?     TypeConv<T, double>(long.Parse(s))
                                        : TypeConv<T, string>(s);
            public static (T,U) Tuple<T,U>() {string[] strs = Strs(); T t = TypeConv<T>(strs[0]); U u = TypeConv<U>(strs[1]); return(t,u);}
            public static (T,U,V) Tuple<T,U,V>() {string[] strs = Strs(); T t = TypeConv<T>(strs[0]); U u = TypeConv<U>(strs[1]); V v = TypeConv<V>(strs[2]); return(t,u,v);}
        }
    }
}