A10337 | Bear and Polynomials

Limak is a little polar bear. He doesn't have many toys and thus he often plays with polynomials.

He considers a polynomial valid if its degree is $n$ and its coefficients are integers not exceeding $k$ by the absolute value. More formally:

Let $a_{0},a_{1},...,a_{n}$ denote the coefficients, so . Then, a polynomial $P(x)$ is valid if all the following conditions are satisfied:

- $a_{i}$ is integer for every $i$ ;
- $|a_{i}|<=k$ for every $i$ ;
- $a_{n}≠0$ .

Limak has recently got a valid polynomial $P$ with coefficients $a_{0},a_{1},a_{2},...,a_{n}$ . He noticed that $P(2)≠0$ and he wants to change it. He is going to change one coefficient to get a valid polynomial $Q$ of degree $n$ that $Q(2)=0$ . Count the number of ways to do so. You should count two ways as a distinct if coefficients of target polynoms differ.

The first line contains two integers $n$ and $k$ ( $1<=n<=200000,1<=k<=10^{9}$ ) — the degree of the polynomial and the limit for absolute values of coefficients.

The second line contains $n+1$ integers $a_{0},a_{1},...,a_{n}$ ( $|a_{i}|<=k,a_{n}≠0$ ) — describing a valid polynomial . It's guaranteed that $P(2)≠0$ .

Print the number of ways to change one coefficient to get a valid polynomial $Q$ that $Q(2)=0$ .