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Points: 10 (partial)
Time limit: 1.0s
Memory limit: 512M

Problem types

These problems are from the atcoder DP contest, and were transferred onto DMOJ. All problem statements were made by several atcoder users. As there is no access to the test data, all data is randomly generated. If there are issues with the statement or data, please contact Rimuru or Ninjaclasher on slack.

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 \le N \le 3000
  • 1 \le a_i \le 10^9

Input Specification

The first line will contain the integer N.

The next line will contain N integers, a_1, a_2, \ldots, a_N.

Output Specification

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

Explanation For Sample 1

The game proceeds as follows when the two players play optimally (the element being removed is written bold):

  • Taro: (10, 80, 90, 30) \rightarrow (10, 80, 90)
  • Jiro: (10, 80, 90) \rightarrow (10, 80)
  • Taro: (10, 80) \rightarrow (10)
  • Jiro: (10) \rightarrow ()

Here, X = 30 + 80 = 110 and Y = 90 + 10 = 100.

Sample Input 2

3
10 100 10

Sample Output 2

-80

Explanation For Sample 2

The game proceeds, for example, as follows when the two players play optimally:

  • Taro: (10, 100, 10) \rightarrow (100, 10)
  • Jiro: (100, 10) \rightarrow (10)
  • Taro: (10) \rightarrow ()

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

Explanation For Sample 4

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

Explanation For Sample 5

The game proceeds, for example, as follows when the two players play optimally:

  • Taro: (4, 2, 9, 7, 1, 5) \rightarrow (4, 2, 9, 7, 1)
  • Jiro: (4, 2, 9, 7, 1) \rightarrow (2, 9, 7, 1)
  • Taro: (2, 9, 7, 1) \rightarrow (2, 9, 7)
  • Jiro: (2, 9, 7) \rightarrow (2, 9)
  • Taro: (2, 9) \rightarrow (2)
  • Jiro: (2) \rightarrow ()

Here, X = 5 + 1 + 9 = 15, and Y = 4 + 7 + 2 = 13.


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