Compound Poisson With Geometric Batch Size

Let $N\sim\mathrm{Poisson}(2)$, and conditional on $N$, let \[ S=\sum_{i=1}^N B_i, \] where the $B_i$ are i.i.d. geometric-on-\{1,2,\dots\} with parameter $1/2$, so $P(B_i=k)=2^{-k}$. Find the PGF of $S$ and compute $E[S]$.