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.

4
10 80 90 30

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 $Y = 90 + 10 = 100$.

3
10 100 10

-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 $Y = 100$.

1
10

10

10
1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000 1

4999999995

The answer may not fit into a 32-bit integer type.

6
4 2 9 7 1 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 $Y = 4 + 7 + 2 = 13$.