A Moore sequential circuit has two inputs $($ X and Y $)$ and three outputs $,$ and Z$0).$ The outputs are a $1\text{'}$s complement number specifying the number of successive times X and Y have been equal or not equal as follows: In decimal, the outputs are $1,2$ and $3,$ if X and Y have been equal for one time, two successive times, and three or more successive times, and the outputs are, $-1,-2,$ and $-3$ if X and Y have been not equal for one time, two successive times, and three or more successive times. Initially, the outputs are all $0.$

$($a$)$ Construct a state diagram for the circuit.

$($b$)$ Encode the states using a one$-$hot state assignment and derive the D flip$-$flop input equations and the output equations.

$($c$)$ Use a minimum number of D flip$-$flops and derive the D flip$-$flop input equations and the output equations.