$($a$)$ Design a polynomial$-$time algorithm that implements this system.

$(5$ points$)$

Specifically, give a polynomial$-$time algorithm that takes the numbers $p_1,p_2,$dots,$p_n,$ and

the lists $L_1,L_2,$dots,$L_k,$ and does one of the following two things:

Return lists $L_1^{\text{'}},L_2^{\text{'}},$dots,$L_k^{\text{'}}$ satisfying conditions $($A$)$ and $($B$)$; or

Report $($correctly$)$ that there is no set of lists $L_1^{\text{'}},L_2^{\text{'}},$dots,$L_k^{\text{'}}$ that satisfy both proper$-$

ties $(A)$ and $(B).$