$16\mathrm{.}23$ A Moore sequential circuit has three inputs $(\backslash ($ X$\_\{2\},$ X$\_\{1\}\backslash ),$ and $\backslash ($ X$\_\{0\}\backslash ))$ that specify a temperature range in a room. The circuit has two outputs $(\backslash ($ I $\backslash )$ and $\backslash ($ D $\backslash ))$ that control a heater for the room; $\backslash ($ I$=1\backslash )$ causes the heater to increase its heat output, and $\backslash ($ D$=1\backslash )$ causes the heater to decrease its heat output. If the temperature range is $0,1$ or $2,$ for three successive clock cycles, the circuit generates $\backslash ($ I$=1\backslash ),$ and conversely if the temperature range is $5,6,$ or $7,$ for three successive clock cycles, the circuit generates $\backslash ($ D$=1\backslash )$; otherwise, $\backslash ($ I$=0\backslash )$ and $\backslash ($ D$=0\backslash ).$

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.