IR-MO 2004 Problem 23

$\mathcal F$ is a family of 3-subsets of set $X$ . Every two distinct elements of $X$ are exactly in $k$ elements of $\mathcal F$ . It is known that there is a partition of $\mathcal F$ to sets $X_1,X_2$ such that each element of $\mathcal F$ has non-empty intersection with both $X_1,X_2$ . Prove that $|X|\leq4$ .