Q$3)$ Design a combinational circuit that accepts a $3-$bit binary input $($ ABC $)$ and generates a $2-$bit output $($ XY $)$ where the output is the total number of $0\text{'}$s in the input.

$($a$)$ Determine the sum$-$of$-$minterms $($SOM$)$ of the circuit.

$($b$)$ Determine the product$-$of$-$maxterms $($POM$)$ of the circuit.

$($c$)$ Optimize the circuit product$-$of$-$sums $($POS$)$ form.

$($d$)$ Implement the circuit with NOR gates only.