Let $S$ be a set such that $|S| = n$. Let $\mathcal{A} = \{A_1, A_2, \dots, A_k\}$ be a collection of distinct subsets of $S$. If every element of $S$ is contained in exactly $m$ subsets of $\mathcal{A}$, what is the relationship between $n, k, m$, and the sizes of the subsets $|A_i|$?