Geometric Number of Geometric Trials

A gambler plays a sequence of rounds. In each round, he flips a biased coin with $P(\text{heads}) = p$ repeatedly until he gets heads; the number of flips in that round is $\operatorname{Geom}(p)$. The number of rounds he plays is itself $\operatorname{Geom}(q)$ (independent of the coin flips), where $0 < q < 1$. Let $S$ be the total number of coin flips across all rounds. Using the memoryless property of the geometric distribution, show that $S \sim \operatorname{Geom}(pq)$ and compute $E[S]$.