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: Accepted)

#include <bits/stdc++.h>

#define FAST_IO ios_base::sync_with_stdio(false), cin.tie(nullptr)
#define FILE_IO freopen("input.txt", "r", stdin), freopen("output.txt", "w", stdout)
#define trav(e, x) for (auto &e : x)
#define pb(x) push_back(x)
#define eb(x...) emplace_back(x)
#define all(x) x.begin(), x.end()
#define sz(x) (int) (x).size()
#define int long long
using namespace std;
typedef pair<int, int> ii;

const int N = 3001;
int a[N];
int dp[N][N][2];

int f(int i, int j, int k) {
    if (i > j) return 0;
    if (dp[i][j][k] > -1) return dp[i][j][k];
    int ans;
    if (k == 1) ans = max(f(i + 1, j, 0) + a[i], f(i, j - 1, 0) + a[j]);
    else ans = min(f(i + 1, j, 1), f(i, j - 1, 1));
    return dp[i][j][k] = ans;
}

signed main() {
    FAST_IO;
    //FILE_IO;

    int n;
    cin >> n;
    int sum = 0;
    for (int i = 0; i < n; i++) {
        cin >> a[i];
        sum += a[i];
    }
    for (int i = 0; i <= n; i++) for (int j = 0; j <= n; j++) dp[i][j][0] = dp[i][j][1] = -1;
    int x = f(0, n - 1, 1);
    int y = sum - x;
    cout << x - y;
}