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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace D
{
    class Program
    {
        static int n, maxK;
        static Dictionary<string, int> memo;

        static void Main(string[] args)
        {

            int[] nk = RArInt();
            n = nk[0];
            maxK = nk[1];
            int[] vs = RArInt();
            memo = new Dictionary<string, int>();

            //int res = 0;
            //for (int i = 0; i < maxK; i++)
            //{
            //    int l = i;
            //    int r = maxK - i;

            //    int lMax = 0;
            //    int rMax = 0;

            //    for (int j = 0; j <= l; j++)
            //    {
            //        lMax = Math.Max(lMax, vs.Take(j).Sum());
            //    }

            //    for (int j = 0; j <= r; j++)
            //    {
            //        rMax = Math.Max(rMax, vs.Skip(vs.Length - j).Take(j).Sum());
            //    }
            //    res = Math.Max(res, lMax + rMax);
            //}
            //Console.WriteLine(res < 0 ? 0 : res);


            LinkedList<int> q = new LinkedList<int>();
            for (int i = 0; i < vs.Length; i++) q.AddLast(vs[i]);
            int res = Dfs(new LinkedList<int>(q), new PriorityQueue<int>(), 0);
            Console.WriteLine(res < 0 ? 0 : res);
        }

        private static int Dfs(LinkedList<int> q, PriorityQueue<int> have, int k)
        {
            int res = have.Buffer.Sum();
            string ky = C(q);
            if (memo.ContainsKey(ky)) return memo[ky];

            if (k < maxK)
            {
                //A
                if (q.Count > 0)
                {
                    LinkedList<int> nq = new LinkedList<int>(q);
                    PriorityQueue<int> nHave = new PriorityQueue<int>(have);
                    nHave.Enqueue(nq.First.Value);
                    nq.RemoveFirst();
                    res = Math.Max(res, Dfs(nq, nHave, k + 1));
                }

                //B
                if (q.Count > 1)
                {
                    LinkedList<int> nq = new LinkedList<int>(q);
                    PriorityQueue<int> nHave = new PriorityQueue<int>(have);
                    nHave.Enqueue(nq.Last.Value);
                    nq.RemoveLast();
                    res = Math.Max(res, Dfs(nq, nHave, k + 1));
                }

                if (have.Count > 0 && have.Peek() < 0)
                {
                    //C
                    LinkedList<int> nq = new LinkedList<int>(q);
                    PriorityQueue<int> nHave = new PriorityQueue<int>(have);
                    nq.AddFirst(nHave.Dequeue());
                    res = Math.Max(res, Dfs(nq, nHave, k + 1));

                    //D
                    if (q.Count() > 0)
                    {
                        LinkedList<int> nq2 = new LinkedList<int>(q);
                        PriorityQueue<int> nHave2 = new PriorityQueue<int>(have);
                        nq2.AddFirst(nHave2.Dequeue());
                        res = Math.Max(res, Dfs(nq2, nHave2, k + 1));
                    }

                }

            }
            if (!memo.ContainsKey(ky)) memo.Add(ky, res);
            return res;
        }

        private static string C(LinkedList<int> q)
        {
            string res = string.Empty;
            foreach (var item in q)
            {
                string s = item.ToString();
                res += item <= 0 ? "-" + s : s;
            }
            return res;
        }

        static string RSt() { return Console.ReadLine(); }
        static int RInt() { return int.Parse(Console.ReadLine().Trim()); }
        static long RLong() { return long.Parse(Console.ReadLine().Trim()); }
        static double RDouble() { return double.Parse(Console.ReadLine()); }
        static string[] RArSt(char sep = ' ') { return Console.ReadLine().Trim().Split(sep); }
        static int[] RArInt(char sep = ' ') { return Array.ConvertAll(Console.ReadLine().Trim().Split(sep), e => int.Parse(e)); }
        static long[] RArLong(char sep = ' ') { return Array.ConvertAll(Console.ReadLine().Trim().Split(sep), e => long.Parse(e)); }
        static double[] RArDouble(char sep = ' ') { return Array.ConvertAll(Console.ReadLine().Trim().Split(sep), e => double.Parse(e)); }
        static string WAr<T>(IEnumerable<T> array, string sep = " ") { return string.Join(sep, array.Select(x => x.ToString()).ToArray()); }
    }

    public class PriorityQueue<T> where T : IComparable<T>

    {

        public List<T> Buffer { get; set; }

        public int Count { get { return Buffer.Count; } }

        public PriorityQueue() { Buffer = new List<T>(); }
        public PriorityQueue(int capacity) { Buffer = new List<T>(capacity); }
        public PriorityQueue(PriorityQueue<T> pq) { Buffer = new List<T>(pq.Buffer); }

        /// <summary>
        /// ヒープ化されている配列リストに新しい要素を追加する。
        /// </summary>
        public void Enqueue(T item)
        {
            int n = Buffer.Count;
            Buffer.Add(item);

            while (n != 0)
            {
                int i = (n - 1) / 2;
                if (Buffer[n].CompareTo(Buffer[i]) * -1 > 0)
                {
                    T tmp = Buffer[n]; Buffer[n] = Buffer[i]; Buffer[i] = tmp;
                }
                n = i;
            }
        }

        /// <summary>
        /// ヒープから最大値を取り出し、削除する。
        /// </summary>
        public T Dequeue()
        {
            T ret = Buffer[0];
            int n = Buffer.Count - 1;
            Buffer[0] = Buffer[n];
            Buffer.RemoveAt(n);

            for (int i = 0, j; (j = 2 * i + 1) < n;)
            {
                if ((j != n - 1) && (Buffer[j].CompareTo(Buffer[j + 1]) * -1 < 0))
                    j++;
                if (Buffer[i].CompareTo(Buffer[j]) * -1 < 0)
                {
                    T tmp = Buffer[j]; Buffer[j] = Buffer[i]; Buffer[i] = tmp;
                }
                i = j;
            }
            return ret;
        }

        /// <summary>
        /// ヒープから最大値を参照する。
        /// </summary>
        public T Peek()
        {
            if (Count == 0) throw new InvalidOperationException();
            return this.Buffer[0];
        }
    }
}