Determine the POS expressions of a $1-$bit PGU and $1-$bit CGU and then draw their circuits using NOT and NOR gates, but combine and use the minimum number of gates. Show each step clearly.

$1)$ Write expressions for Carry$-$generate logic $$g$.$ Use POS form. Realize it using NOT and NOR gates.

$2)$ Write expressions for Carry$-$propagate $$p$.$ Use POS form. Realize it using NOT and NOR gates.

$3)$ Write an expression for Cout $($carry$-$out$)$ using

$($a$)$g$,$P$$ and $$Cin$.$

$($b$)$ Write a truth table for Cout using $3$ input variables: g$,$ p$,$ cin. Write the SOP form.

$($c$)$ Using Karnaugh maps optimize this POS form.

$($d$)$ Realize the logic using the NOT, and NOR gates.