Let $W$ be any non-empty set. We call a subset A of $\mathscr{P}(W)$ a 3-fold filter on $W$ if the following hold:

(i) $W\in A$ and $\varnothing\notin A$;

(ii) If $a\in A$ and $a\subset b$ then $b\in A$;

(iii) If $a,b,c\in A$ then $(a\cap b)\cup(b\cap c)\cup(c\cap a)\in A$.

Now let $A,B$ be two 3-fold filters on $W$. Let $X=\{a\times W\mid a\in A\}$ and $Y=\{W\times b\mid b\in B\}$. Prove that $X\cup Y$ can be extended to a 3-fold filters on $W\times W$, i.e., $\varnothing$ is not contained in the closure of $X\cup Y$ under the operation $(a\cap b)\cup(b\cap c)\cup(c\cap a)$.