Question $8$: $[20$ marks$]($Moderate but Time Consuming$)$

Using T$-$FFs$,$ design a $3-$bit register $[$C$,$B$,$A$]$ with a single input control x $.$ If $x=1,$ the register shifts the value of the memory units to the right. If $x=0,$ the register shifts them to the left. In shift right, $0$ enters C $.$ In shift left, $0$ enters A$.$ For instance: $x=1$:$011001,x=0$:$011110.$

a$.[5$ marks$]$ Draw the state diagram of the circuit

b$.[5$ marks$]$ Derive the state table

c$.[10$ marks$]$ Draw the minimized logic diagram of the circuit