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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include<bits/stdc++.h>
#define ll long long int
using namespace std;

int dp[3001][3001][3];
//1-Taro , 0-Jiro
ll recurse(int l , int r, int turn , vector<ll> &arr)
{
    if(l>r) return 0;
    else
    {
        if(dp[l][r][turn] != -1) return dp[l][r][turn];
        if(turn) dp[l][r][turn] = max(recurse(l+1,r,0,arr)+arr[l] , recurse(l,r-1,0,arr)+arr[r]);
        else dp[l][r][turn] = min(recurse(l+1,r,1,arr),recurse(l,r-1,1,arr));
        return dp[l][r][turn];
    }
    
}

int main(int argc, char const *argv[])
{
    int n;
    ll temp,total= 0;
    cin >> n;
    vector<ll> arr(n,0);
    for(int i=0 ; i < 3000 ; i++)for(int j = 0; j < 3000 ; j++)for(int k =0 ; k <3 ; k++) dp[i][j][k] = -1;
    for(int i =0 ; i < n ; i++)
    {
        cin >> temp;
        arr[i] = temp;
        total+= temp;
    }
    ll x = recurse(0,n-1,1,arr);
    ll y = total - x;
    cout << x-y;
    return 0;
}