complexity theory – Polynomial kernelization for “Set coloring” problem

Given an universum $U$ and a system of its subsets $Fsubseteq mathcal{P}(U)$, given a $kin mathbb{Z}_{geq 0}$. The decision problem is to say whether there exists a $2$-coloring of $U$ s.t. atleast $k$ sets from $F$ contain elements of both colors.

I wish to show that there exists kernel with $2k$ sets and $|U|in O(k^2)$. So far I have these reduction rules:

Rule 1: If there is $xin U$ s.t. there’s no $Win F$ s.t. $xin W$, then we can delete $x$ from U and work with a new instance. $(F,Usetminus{x},k)$.

Rule 2: If $Win F$ is a singleton, then we can remove it and work with $(Fsetminus {W},U,K)$ because such $W$ will never contain two elements of different colors.

Not really sure how to proceed from here. Any hints appreciated.