Problem:
Score : 100 points

Problem Statement

Taro and Jiro will play the following game against each other.

Initially, they are given a sequence a = (a_1, a_2, \ldots, a_N).
Until a becomes empty, the two players perform the following operation alternately, starting from Taro:

Remove the element at the beginning or the end of a. The player earns x points, where x is the removed element.

Let X and Y be Taro's and Jiro's total score at the end of the game, respectively.
Taro tries to maximize X - Y, while Jiro tries to minimize X - Y.

Assuming that the two players play optimally, find the resulting value of X - Y.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the resulting value of X - Y, assuming that the two players play optimally.

Sample Input 1

4
10 80 90 30

Sample Output 1

10

The game proceeds as follows when the two players play optimally (the element being removed is written bold):

Taro: (10, 80, 90, 30) → (10, 80, 90)

Jiro: (10, 80, 90) → (10, 80)

Taro: (10, 80) → (10)

Jiro: (10) → ()

Here, X = 30 + 80 = 110 and Y = 90 + 10 = 100.

Sample Input 2

3
10 100 10

Sample Output 2

-80

The game proceeds, for example, as follows when the two players play optimally:

Taro: (10, 100, 10) → (100, 10)

Jiro: (100, 10) → (10)

Taro: (10) → ()

Here, X = 10 + 10 = 20 and Y = 100.

Sample Input 3

1
10

Sample Output 3

10

Sample Input 4

10
1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000 1

Sample Output 4

4999999995

The answer may not fit into a 32-bit integer type.

Sample Input 5

6
4 2 9 7 1 5

Sample Output 5

2

The game proceeds, for example, as follows when the two players play optimally:

Taro: (4, 2, 9, 7, 1, 5) → (4, 2, 9, 7, 1)

Jiro: (4, 2, 9, 7, 1) → (2, 9, 7, 1)

Taro: (2, 9, 7, 1) → (2, 9, 7)

Jiro: (2, 9, 7) → (2, 9)

Taro: (2, 9) → (2)

Jiro: (2) → ()

Here, X = 5 + 1 + 9 = 15 and Y = 4 + 7 + 2 = 13.

Solution in C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using static System.Console;

namespace dp
{
    class Program
    {
        static int[] ReadNumList()
        {
            var line = ReadLine().Split(' ');
            if (line.Length == 1 && string.IsNullOrEmpty(line[0]))
            {
                return new int[0];
            }
            var numList = new int[line.Length];
            for (var i = 0; i < line.Length; ++i)
            {
                numList[i] = int.Parse(line[i]);
            }
            return numList;
        }
        static void Main()
        {
            var n = int.Parse(ReadLine());
            var aList = ReadNumList();

            var dp = new long[n + 1][];
            for (var i = 0; i < dp.Length; ++i)
            {
                dp[i] = new long[n + 1];
            }

            WriteLine(Search(dp, 0, n, aList));
        }
        static long Search(long[][] dp, int first, int last, int[] numList)
        {
            var isSente = (dp.Length - last + first) % 2 == 1;
            var result = 0L;
            if (last - first == 1)
            {
                if (isSente)
                {
                    result = numList[first];
                }
                else
                {
                    result = -numList[first];
                }
            }
            else
            {
                if (isSente)
                {
                    result = Math.Max(
                        Search(dp, first + 1, last, numList) + numList[first],
                        Search(dp, first, last - 1, numList) + numList[last - 1]);
                }
                else
                {
                    result = Math.Min(
                        Search(dp, first + 1, last, numList) - numList[first],
                        Search(dp, first, last - 1, numList) - numList[last - 1]);
                }
            }
            dp[first][last] = result;
            return result;
        }
    }
}