$1)$ A sequential logic circuit consists of two states $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ and has input $\backslash ($ C $\backslash )$ and has output signal $\backslash ($ F $\backslash ).$ It uses D$-$flip flops. For the sequential logic circuit, whose state table is shown below, find its simplest state equations, implement the corresponding circuit schematics, and its state diagram:

$($a$)$ Find the simplest state equations of this circuit $(35$ points$).$

$($b$)$ Implement the sequential logic circuit schematics that implement the equations above $(35$ points$).$

$($c$)$ Design the state diagram for this circuit $(30$ points$).$