Poisson-Exponential Sum with Shared Rate: Double Tower and Eve's Law

Let $Z \sim \operatorname{Uniform}(1, 3)$. Given $Z = z$, let $N \mid Z \sim \operatorname{Poisson}(z)$, and given $(N, Z)$, let $X_1, \ldots, X_N$ be i.i.d.\ $\operatorname{Exp}(z)$ (rate $z$). Set $S = X_1 + \cdots + X_N$ (with $S = 0$ when $N = 0$).

Using iterated tower properties and Eve's law, find $E[S]$ and $\operatorname{Var}(S)$.