$(15\%)$ From equation to state graph.

A sequential circuit has three D flip$-$llops, A$,$ B$,$ C$,$ one input $x_{,\text{a}\text{n}\text{d}}$ one output $Z.$ The flip$-$flop input equations are described as follows:

$A^{**}=D_1=A^{\text{'}}B{C}^{\text{'}}+x{A}^{\text{'}}{B}^{\text{'}}C+x^{\text{'}}A{B}^{\text{'}}{C}^{\text{'}}$

$B^{**}=D_1=x{A}^{\text{'}}C+x{A}^{\text{'}}{C}^{\text{'}}$

$C^{**}=D_C=x{A}^{\text{'}}{A}^{\text{'}}+x{A}^{\text{'}}C+A^{\text{'}}{B}^{\text{'}}C$

$($a$)3\%$ Please derive the state table for the design. $($assume $5_0=$

$($:$110,5,s^{\text{'}}=111\})$:$\}$

$($b$)4\%$ Please draw the state graph if the output equation is: $Zx{A}^{\text{'}}BC+xA{B}^{\text{'}}{C}^{\text{'}}+xA{A}^{\text{'}}BC+xAB{B}^{\text{'}}{C}^{\text{'}}$

$($c$)[4\%]$ Please draw the state graph if the output equation is: $Z=A{B}^{\text{'}}{C}^{\text{'}}+A^{\text{'}}BC$

$($d$)[4\%]$ Continue on $($c$),$ can you use only $2$ state variables to implement a state machine which have the same function? if so$,$ please show how you eliminate the redundant states, and draw the state eraph with minimum number of states.