Is it possible to construct a family of sets $\{A_{ij}\}_{i,j=1}^\infty$ and $\{B_{ij}\}_{i,j=1}^\infty$ such that:

$(1)$For any positive integer $i\geq1$,$A_{ij}\searrow\emptyset$ and $B_{ij}\searrow\emptyset$ as $j\to\infty$;

$(2)\bigcup_{i=1}^\infty A_{i1}=\bigcup_{i=1}^\infty B_{i1}$;

$(3)A_{11},A_{21},A_{31}...$are pairwise disjoint and $B_{11},B_{21},B_{31}...$are pairwise disjoint;

$(4)$For any positive integers $i$,there exist positive integers $j_1,j_2$ and sequences of positive integers $\{i_k\}_{k=1}^\infty,\{j_k\}_{k=1}^\infty$ such that $A_{ij_1}\bigcap (\bigcup_{k=1}^\infty B_{ki_k})=\emptyset$ and $B_{ij_2}\bigcap (\bigcup _{k=1}^\infty A_{kj_k})=\emptyset$;

$(5)$For any sequences of positive integers $\{i_k\}_{k=1}^\infty$ and $\{j_k\}_{k=1}^\infty$, $(\bigcup_{k=1}^\infty A_{ki_k})\bigcap (\bigcup_{k=1}^\infty B_{kj_k})\neq\emptyset$.