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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define int long long
#define y1 zzz
#define fi first
#define se second
#define PII pair<int,int>
#define PII2 pair<int,PII >
#define PII3 pair<PII,int >
#define PII4 pair<PII,PII >
#define all(x) (x).begin(),(x).end()
#define pb push_back
int dp[3010][3010][2],a[3010],n;
int f(int l,int r,int state)
{
    if(l==r)
    {
        if(state)return -a[l];
        return a[l];
    }
    if(dp[l][r][state]!=-1e18)return dp[l][r][state];
    if(state)return dp[l][r][state]=min(-a[l]+f(l+1,r,1-state),-a[r]+f(l,r-1,1-state));
    return dp[l][r][state]=max(a[l]+f(l+1,r,1-state),a[r]+f(l,r-1,1-state));
}
void solve()
{
    cin>>n;
    for(int i=1;i<=n;i++)cin>>a[i];
    for(int i=1;i<=n;i++)
        for(int j=1;j<=n;j++)
            dp[i][j][0]=dp[i][j][1]=-1e18;
    cout<<f(1,n,0);
}
int32_t main()
{
    ios_base::sync_with_stdio(0);
    cin.tie(0);cout.tie(0);
    //freopen("XORQUERY.inp","r",stdin);
    //freopen("XORQUERY.out","w",stdout);
    //int t;cin>>t;while(t--)
    solve();
    return 0;
}