Question $4$: Functional Design $(30$ points$)$

Give the functional design of a logic circuit which takes two $4-$bit unsigned numbers A and $B$ as input and outputs a $4-$bit unsigned number $Z$ such that $Z=\frac{A+B}{2}$ when $A+B>15.$ In all other cases, $Z=A-B$ if $A>B$ and $Z=B-$A otherwise.

You may use the following building blocks:

$4-$bit adders $($inputs and output are both $4-$bits$)$ with carry in and carry out.

$4-$bit unsigned magnitude comparators with $3$ outputs L$,$ E$,$ and G $($respectively Less than, Equal to$,$ and Greater than$).$

$4-$bit $2-$to$-1$ multiplexers.

$4-$bit $2\text{'}$s complement circuits $($i$.$e$.$: input $x,$ output $2\text{'}$s complement of $x).$

AND, OR$,$ XOR, and NOT gates if needed.

In case you need any other building block you must provide its detailed circuit.

Explain your design.