Compound Poisson with Exponential Severities

Claims arrive according to $N\sim\mathrm{Poisson}(\lambda)$. Claim sizes $X_1,X_2,\dots$ are i.i.d. $\mathrm{Exponential}(\beta)$ with rate $\beta$, independent of $N$. Let $S=\sum_{i=1}^N X_i$. Derive the MGF of $S$, and compute $E[S]$ and $\mathrm{Var}(S)$.