A Boolean function $Y$ of four input variables is given as

$Y(A,B,C,D)={\displaystyle \underset{m}{\overset{?}{}}}(1,2,3,5,9,10,11,12)$

Don't care: ${\displaystyle \underset{m}{\overset{?}{}}}(8,13,15)$

$($a$)$ List the minterm numbers in binary values. The first one was done for you.

$1={0001}_2$

$($b$)$ By Quine$-$McCluskey tabular method, find all the prime implicants of Y$.$

$($c$)$ Determine all the essential prime implicants of Y$.$

$($d$)$ Find the simplest Boolean expression of Y$.$