A Geometric Number of Exponential Stages

A task takes a geometric number of stages: $N$ has support $\{1,2,\dots\}$ with $P(N=n)=p(1-p)^{n-1}$. Each stage duration is i.i.d. $\mathrm{Exponential}(\beta)$, independent of $N$. Let $T=\sum_{i=1}^N X_i$. Use MGFs to identify the law of $T$.