Design a $4-$state synchronous counter with a Reverse input $($R$),$ the counter produces the sequence: $0,1,2,3$ then repeats the count when R$=0.$ When R$=1$ the count sequence is $3,2,1,0,$ then repeats. The counter is to be implemented using JK flip$-$flops. The output Z indicates whether the counter is counting UP or DOWN. a$)$ Complete the state transition table below b$)$ Derive the input equations for each JK flip$-$flop in minimal sum$-$of$-$products form c$)$ Show the equation for the output Z