Let $\mathcal{F} = \{A_1, A_2, \dots, A_k\}$ be a family of $k$ distinct non-empty subsets of a universal set $U = \{1, 2, \dots, n\}$. If for any two distinct sets $A_i, A_j \in \mathcal{F}$, their intersection is empty (i.e., $A_i \cap A_j = \emptyset$ for $i \ne j$), and their union is $U$ (i.e., $A_i \cup A_j = U$ for $i \ne j$), what can be concluded about the cardinality of $k$ and the sets themselves?