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.

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.

Constraints

• All values in input are integers.
• $1 \leq N \leq 3000$
• $1 \leq a_i \leq 10^9$

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.

见题面 Input 部分

见题面 Output 部分