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: Runtime Error)

#include<bits/stdc++.h> 
using namespace std;

#define int long long

int solve(vector<int>& coins, int l, int r,vector<vector<int>> &dp){
	if (l>r) return 0;
	if (dp[l][r]!=-1) return dp[l][r];
	return max(coins[l]+min(dp[l+2][r]!=-1?dp[l+2][r]:dp[l+2][r]=solve(coins,l+2,r,dp),dp[l+1][r-1]!=-1?dp[l+1][r-1]:dp[l+1][r-1]=solve(coins,l+1,r-1,dp)),
		   	   coins[r]+min(dp[l+1][r-1]!=-1?dp[l+1][r-1]:dp[l+1][r-1]=solve(coins,l+1,r-1,dp),dp[l][r-2]!=-1?dp[l][r-2]:dp[l][r-2]=solve(coins,l,r-2,dp)));
}

signed main() {
    // freopen("input.txt", "r", stdin);
    // freopen("output.txt", "w", stdout);
	int N;
	cin>>N;
	int sum=0;
	vector<int> coins(N);
	for (int i=0;i<N;i++){
		cin>>coins[i];
		sum+=coins[i];
	}
	vector<vector<int>> dp(N+1,vector<int>(N+1,-1));
	int X = solve(coins,0,N-1,dp);
	int Y = sum-X;
	cout<<X-Y<<endl;
    return 0; 
}