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 Pascal (FPC 2.6.2) (judge verdict: Wrong Answer)

uses crt;
{
const
        fi = 'test.inp';
        fo = 'test.out';
}
var
        f: Text;
        a,b: array [0..4000] of int64;
//        dp: array [0..1000, 0..1000] of longint;
        i,j,n: longint;
        x,y: int64;

begin
//        assign (f, fi); reset (F);
        readln (n);
        for i:= 1 to n do
        begin
                read ( a[i]);

        end;
    //    close (f);

      //  assign (F, fo); rewrite (f);
        j:=0;
        for i:=1 to n do
        begin
                inc(j);
                b[j]:= a[i];

                if  ( j - 3 >= 0 ) then

                if ( b[j-1] > b[j] ) and (b[j-1] > b[j-2]) then
                begin
                        b[j-2]:= b[j-2] +b[j] - b[j-1];
                        j:= j-2;
                end;

        end;

        i:=1;

        while i <= j do
        begin
                if b[i] > b[j] then
                begin
                        x:= x + b[i]; inc(i);
                end else
                begin
                        x:= x + b[j]; dec(j);
                end;

                if i > j then break;
                if b[i] > b[j] then
                begin
                        y:= y + b[i]; inc(i);
                end else
                begin
                        y:= y + b[j]; dec(j);
                end;
        end;

     //   for i:=1 to n do write (f, b[i], ' ');   writeln (f);
        write (x- y);
        readln;
     //   close (f);
end.