Problem:
Score : 400 points

Problem Statement

Your friend gave you a dequeue D as a birthday present.

D is a horizontal cylinder that contains a row of N jewels.

The values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.

In the beginning, you have no jewel in your hands.

You can perform at most K operations on D, chosen from the following, at most K times (possibly zero):

Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.

Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.

Find the maximum possible sum of the values of jewels in your hands after the operations.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 100

-10^7 \leq V_i \leq 10^7

Input

Input is given from Standard Input in the following format:

N K
V_1 V_2 ... V_N

Output

Print the maximum possible sum of the values of jewels in your hands after the operations.

Sample Input 1

6 4
-10 8 2 1 2 6

Sample Output 1

14

After the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.

Do operation A. You take out the jewel of value -10 from the left end of D.

Do operation B. You take out the jewel of value 6 from the right end of D.

Do operation A. You take out the jewel of value 8 from the left end of D.

Do operation D. You insert the jewel of value -10 to the right end of D.

Sample Input 2

6 4
-6 -100 50 -2 -5 -3

Sample Output 2

44

Sample Input 3

6 3
-6 -100 50 -2 -5 -3

Sample Output 3

0

It is optimal to do no operation.

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;

namespace Sample
{
    public class Program
    {
        public static void Main(string[] args)
        {
            using (var sc = new SetConsole()) { Solve(); }
        }

        public static void Solve()
        {
            var nk = GetArray<int>();
            var n = nk[0];
            var k = nk[1];
            var vs = GetArray<int>().ToList();
            var count = Min(n, k);
            var results = new List<int> { 0 };
            for (int i = 1; i <= count; i++)
            {
                var thisCount = i;
                var takeList = Search(vs, thisCount);
                var minuss = new Queue<int>(takeList.Where(t => 0 > t).OrderBy(o => o));
                while (count != takeList.Count && minuss.Any())
                {
                    takeList.Add(-1 * minuss.Dequeue());
                    thisCount--;
                }
                results.Add(takeList.Sum());
            }

            CWrite(results.Max());
        }

        public static List<int> Search(List<int> vs, int count)
        {
            if (count == 0 || !vs.Any()) { return new List<int>(); }

            var tmp1 = new List<int>(vs);
            var tmp1Value = tmp1[0];
            tmp1.RemoveAt(0);
            var result1 = Search(tmp1, count - 1);
            result1.Add(tmp1Value);

            var tmp2 = new List<int>(vs);
            var tmp2Value = tmp2[tmp2.Count - 1];
            tmp2.RemoveAt(tmp2.Count - 1);
            var result2 = Search(tmp2, count - 1);
            result2.Add(tmp2Value);

            if (result1.Where(r => r >= 0).Sum() > result2.Where(r => r >= 0).Sum()) { return result1; }
            return result2;
        }


        static void CWrite<T>(T str) => Console.WriteLine(str);
        static string GetString() => Console.ReadLine();
        static T[] GetArray<T>()
        {
            var str = Console.ReadLine();
            if (str == null) { throw new ArgumentNullException("標準入力がnullです 解答がおかしいはずです"); }

            var t = typeof(T);
            if (t == typeof(string))
            {
                return (T[])(object)str.Split();
            }
            if (t == typeof(int))
            {
                return (T[])(object)str.ToNumArray<T>();
            }
            if (t == typeof(long))
            {
                return (T[])(object)str.ToNumArray<T>();
            }
            if (t == typeof(double))
            {
                return (T[])(object)str.ToNumArray<T>();
            }
            throw new NotSupportedException($"{t} is not supported.");
        }
    }



    class SetConsole : IDisposable
    {
        readonly StreamWriter sw = new StreamWriter(Console.OpenStandardOutput());
        public SetConsole()
        {
            sw.AutoFlush = false;
            Console.SetOut(sw);
        }
        public void Dispose()
        {
            Console.Out.Flush();
            sw.AutoFlush = true;
            Console.SetOut(sw);
        }
    }

    static class ExtentionsLibrary
    {
        public static T[] CopyArray<T>(this T[] array)
        {
            var newArray = new T[array.Length];
            Array.Copy(array, newArray, array.Length);
            return newArray;
        }
        public static T[,] CopyArray<T>(this T[,] array)
        {
            var firstDimentionLength = array.GetLength(0);
            var secondDimentionLength = array.GetLength(1);
            var newDArray = new T[firstDimentionLength, secondDimentionLength];
            Array.Copy(array, newDArray, firstDimentionLength * secondDimentionLength);
            return newDArray;
        }
        public static T[] ToNumArray<T>(this string str)
        {
            var t = typeof(T);
            if (t == typeof(int))
            {
                return (T[])(object)str.Split().Select(int.Parse).ToArray();
            }
            if (t == typeof(long))
            {
                return (T[])(object)str.Split().Select(long.Parse).ToArray();
            }
            if (t == typeof(double))
            {
                return (T[])(object)str.Split().Select(double.Parse).ToArray();
            }
            throw new NotSupportedException();
        }
        public static int ToInt(this string str) => int.Parse(str);
        public static int ToInt(this char chr) => int.Parse(chr.ToString());
        public static long ToLong(this string str) => long.Parse(str);
        public static DateTime ToDateTime(this string str) => DateTime.Parse(str);
    }
}