Consider the following two equivalent logic equations for $Z.$

$Z=((A*B)+(A*C)+(B*C))\frac{*}{b}ar((A*B*C))$

$Z=(A*B\frac{*}{b}ar(C))+(A\frac{*}{b}ar(B)*C)+(\frac{?}{\text{b}\text{a}\text{r}}(A)*B*C)$

Draw a schematic diagram for each equation with minimum number of $2-$input gates.

Answer which equation is more efficient in terms of the number of $2-$input gates.

Note that the negator is not counted for this.

Prove that the two equations for $Z($shown in #$3)$ are equivalent by using DeMorgan's laws

and other basic laws $($if needed$).$