Let $A$ and $B$ be two sets. We define the set $S = P(A \cup B) \setminus (P(A) \cup P(B))$, where $P(X)$ denotes the power set of $X$. If $|A| = 4$, $|B| = 5$, and $|A \cap B| = 2$, find the cardinality of the set $S$.