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)

// greedi
#include <iostream>
using namespace std;

const int MAXN = 3000;
int arr[MAXN];

int main () {
	int n;
	cin >> n;
	for (int i = 0; i < n; i ++)
		cin >> arr[i];
	long long ans[2]= {0, 0};
	int pointerleft = 0; 
	int pointerright = n - 1;
	int round = 0;
	while (pointerleft <= pointerright) {
		if (pointerleft == pointerright) {
			ans[round] += arr[pointerleft];
			pointerleft ++;
			continue;
		}
		int bleft = arr[pointerleft + 1] - arr[pointerleft]; // behinegi left
		int bright = arr[pointerright - 1] - arr[pointerright]; // behinegi right
		if (bright == bleft) {
			if (arr[pointerright] > arr[pointerleft]) {
				ans[round] += arr[pointerright];
				pointerright --;
			}
			else {
				ans[round] += arr[pointerleft];
				pointerleft ++;
			}
			round += 1;
			round %= 2;
			continue;
		}
		if (bright < bleft) {
			ans[round] += arr[pointerright];
			pointerright --;
			round ++;
			round %= 2;
			continue;
		}
		ans[round] += arr[pointerleft];
		pointerleft ++;
		round ++;
		round %= 2;
	}
	cout << ans[0] - ans[1] << endl;
	return 0;
}