Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Consider a subset $A \subseteq S$. We define two properties for $A$: Property $P_1$: $A$ contains at least one prime number. Property $P_2$: The sum of the elements in $A$ is an even number. How many non-empty subsets $A \subseteq S$ satisfy both Property $P_1$ and Property $P_2$?