Two-Layer Tower with Bernoulli-Switched Exponential Rate

Let $Z \sim \operatorname{Bernoulli}(1/2)$. Given $Z = 1$, let $Y \sim \operatorname{Exp}(1)$; given $Z = 0$, let $Y \sim \operatorname{Exp}(2)$ (rate parametrisation). Given $Y = y$, let $X \sim \operatorname{Poisson}(y)$. Using iterated applications of the tower property and Eve's law, find $E[X]$ and $\operatorname{Var}(X)$.