2011/finals/alien_game.md

Aliens on the Unknown planet have a tradition of playing a game called Loiten.
It is played by two players who alternate turns. There are **N** buckets with
apples standing in one line in front of the players. They are numbered from
left to right with integers starting from 1.

In one turn a player can select one of the buckets, which is not the first and
not the last and has a positive number of apples in it, and move all of that
bucket's apples to the bucket adjacent to the left and at the same time move
all of them to the bucket adjacent to the right. That's right, the number of
apples can be negative as it is a really strange planet. Thus, if there are 3
consecutive buckets with the number of apples being **x**, **y**, **z**, then
you can perform the move if **y** is greater than zero. The resulting capacity
of the buckets will be as follows: **x+y**, **-y**, **z+y**. The first player
who can't make a move loses.

You happen to know one of the aliens from the Unknown planet, named Popo. He
is a very good Loiten player, and has reached the Loiten Finals. On the day
prior to the game, he found out the number of apples in each of the buckets.
Unfortunately, his memory is not that good, and he can't remember the number
of apples in the **P**-th bucket. He just remembers that it is a number with
absolute value not greater than **F**.

Now, he is asking you to help him to calculate his chances. The players at the
Finals are so good that they only make optimal moves to maximize their chance
of winning. If neither player can win, the game is considered a draw. You are
to find the number of possible apple counts for the bucket with an unknown
number of apples where Popo will win. Popo is also sure that he is the one to
make the first turn.

## Input

The first line of the input file consists of a single number **T**, the number
of test cases. Each test case begins with a line containing two integers
**N**, the number of buckets and **P**, the number of the bucket with the
unknown amount of apples. It is followed by a line containing **N** integers,
the numbers of apples in the corresponding buckets. The **P**th number on this
line is the positive integer **F** and corresponds to the constraint on the
number of apples in the **P**-th bucket.  
  

## Output

Output **T** lines, with the answer to each test case on a single line, the
number of possible values for unknown bucket.  
  

## Constraints

**T** = 50  
1≤ **P** ≤ **N** ≤ 2,000.  
1≤ **F** ≤ 1,000,000,000,000.  
The number of apples in each bucket at the start of the game has an absolute
value not greater than 1,000,000,000,000.