Entropy bound for pairwise-union tuples

Let $A_1,\dots ,A_m$ be random subsets of a common ground set. If every pairwise union $A_i\cup A_j$ takes values in a fixed family $\mathcal{A}$ of sets of cardinality at most $k$, then Shearer's inequality and the fact that a pair of bits with prescribed OR has at most three possibilities give