A Geometric Parent Count With Even-Sized Batches

Let $N$ have the geometric law on $\{0,1,2,\dots\}$ with $P(N=n)=\frac13\left(\frac23\right)^n$. Conditional on $N$, let \[ S=\sum_{i=1}^N B_i, \] where $P(B_i=0)=P(B_i=2)=1/2$. Compute $P(S=4)$ using the PGF.