Distribution Theory — deriving a geometric marginal from a joint pmf

Let $X$ and $Y$ be jointly distributed with joint probability mass function

$$P(x,y)=\frac{1}{2^{y+2}(y+1)}\left(\frac{2y+1}{2y+2}\right)^x; \quad x,y=0,1,2,…$$

With reference to the above, consider the following statements:

$$\text{I. The marginal distribution of } Y \text{ is geometric distribution.}$$

$$\text{II. } \operatorname{Var}(Y)=1$$

Which of the statements given above is/are correct?