$(25$ pts$)$ For the following function, find minimal solution in terms of Product$-$of$-$Sums expression,

$F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}m(0,2,5,7,10,13,15)+{\displaystyle \underset{?}{\overset{?}{}}}d(4,8,14)$

a$)$ Using Karnaugh map method.

b$)$ Using AND, OR$,$ and NOT gates realize the minimal function. Use uncomplemented inputs only.

c$)$ Implement the minimal function using NOR gates only. Use uncomplemented inputs only.