KöMaL 2014 A625

Let $n\ge2$, and let $\mathcal{S}$ be a family of some subsets of $\{1,2,\ldots,n\}$ with the property that $|A\cup B\cup C\cup D|\le n-2$ for all $A,B,C,D\in\mathcal{S}$ . Show that $|\mathcal{S}|\le 2^{n-2}$.