Let $S = \{1, 2, \dots, 8\}$. For any subset $X \subseteq S$, let $f(X) = \sum_{x \in X} x$ denote the sum of its elements (with $f(\emptyset) = 0$). Let us define two collections of subsets: $\mathcal{A} = \left\{ A \in P(S) : f(A) \text{ is even} \right\}$ $\mathcal{B} = \left\{ B \in P(S) : |B \cap \{1, 2, 3, 4\}| \ge 3 \right\}$ Determine the cardinality of the intersection $\mathcal{A} \cap \mathcal{B}$.