A15145 | Dog Walking

You are walking with your dog, and now you are at the promenade. The promenade can be represented as an infinite line. Initially, you are in the point $0$ with your dog.

You decided to give some freedom to your dog, so you untied her and let her run for a while. Also, you watched what your dog is doing, so you have some writings about how she ran. During the $i$ -th minute, the dog position changed from her previous position by the value $a_i$ (it means, that the dog ran for $a_i$ meters during the $i$ -th minute). If $a_i$ is positive, the dog ran $a_i$ meters to the right, otherwise (if $a_i$ is negative) she ran $a_i$ meters to the left.

During some minutes, you were chatting with your friend, so you don't have writings about your dog movement during these minutes. These values $a_i$ equal zero.

You want your dog to return to you after the end of the walk, so the destination point of the dog after $n$ minutes should be $0$ .

Now you are wondering: what is the maximum possible number of different integer points of the line your dog could visit on her way, if you replace every $0$ with some integer from $-k$ to $k$ (and your dog should return to $0$ after the walk)? The dog visits an integer point if she runs through that point or reaches in it at the end of any minute. Point $0$ is always visited by the dog, since she is initially there.

If the dog cannot return to the point $0$ after $n$ minutes regardless of the integers you place, print -1.

The first line of the input contains two integers $n$ and $k$ ( $1 \le n \le 3000; 1 \le k \le 10^9$ ) — the number of minutes and the maximum possible speed of your dog during the minutes without records.

The second line of the input contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $-10^9 \le a_i \le 10^9$ ), where $a_i$ is the number of meters your dog ran during the $i$ -th minutes (to the left if $a_i$ is negative, to the right otherwise). If $a_i = 0$ then this value is unknown and can be replaced with any integer from the range $[-k; k]$ .

If the dog cannot return to the point $0$ after $n$ minutes regardless of the set of integers you place, print -1. Otherwise, print one integer — the maximum number of different integer points your dog could visit if you fill all the unknown values optimally and the dog will return to the point $0$ at the end of the walk.