Poisson Number of Geometric Ticket Generators

Let $N\sim\mathrm{Poisson}(3)$. Conditional on $N$, let \[ S=\sum_{i=1}^N B_i, \] where each $B_i$ is geometric on $\{0,1,2,\dots\}$ with success parameter $p=1/4$, so $P(B_i=k)=\frac14(\frac34)^k$. Find the PGF of $S$ and compute $E[S]$.