CN-TST 2015 Problem 14

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$ , satisying:

(1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$ .

Show that one can select $[\frac{3k}{7}]$ sets from $A_1,...,A_k$ such that their union is $X$ .