$1.$ Consider the following K map, on variables $\backslash (\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$B$\},\backslash$mathrm$\{$C$\}\backslash ),$ and D $.$

$($a$)(2$ marks$)$ List all prime implicants, and indicate which ones are essential prime implicants.

$($b$)(2$ marks$)$ Using only prime implicants, find at least two different ways of writing the function in fully reduced sum$-$of$-$products form.

$($c$)(1$ mark$)$ In the fully reduced sum$-$of$-$products expression, why are there no implicants other than prime implicants?