$9.$ Consider the circuit shown below.

$(15$pts$.)$

a$.$ Determine expressions for Z and $\backslash (\backslash$overline$\{\backslash$mathrm$\{$Z$\}\}\backslash )$ useful for hazard analysis. $(3$pts$.)$ b$.$ For single input changes, there exist static$-0$ hazards for the following case$($s$)$: $($State input conditions if any$)$

$(3$pts$.)$

c$.$ For single input changes, there exist static$-1$ hazards for the following case$($s$)$: $($State input conditions if any$)(3$pts$.)$ d$.$ For single input changes, there exist dynamic hazards for the following case$($s$)$: $($State input conditions if any$)(3$pts$.)$

e$.$ Design a hazard$-$free product of sums circuit which realizes this same function.