CIIM 2014 Problem 3

Given $n\geq2$ , let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$ , it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$ . Prove that $|\mathcal{A}| \leq 2^{n-2}.$