$1)$ Complete the following Boolean simplifications using K$-$maps and Boolean algebraic identities and draw the simplest circuits for these expressions. Each part is worth $25$ points. $(100$ points$)$

$($a$)$ Find the simplest F$($A$,$ B$,$ C$,$ D$)$ and draw its simplest circuit.

$($b$)$ Find the simplest F$($X$,$ Y$,$ Z$,$ W$).$ Draw its simplest two$-$stage sum$-$of$-$products circuit.

$($c$)$ Find the simplest $\backslash ($ F$($X$,$ Y$,$ Z$,$ W$)\backslash )$ and draw its simplest circuit only using NAND gates.

$($d$)$ Find the simplest F$($X$,$ Y$,$ Z$,$ W$)$ and draw its simplest circuit only using NOR gates.