$($g$)$ Implement $F$ with a $8-1$ multiplexer

$2.($a$)$ Draw the state diagram for a FSM with states xmod$6,$ and the transitions are $x2x$ mod $6$ or $x3$xmod$6$ based on some binary input I, the outputs are xmod$3$ and $x+1$mod$3$ respectively.

$($b$)$ Encode the states of the FSM by bits. List your encoding in a table or show the encoding on your diagram.

$($c$)$ Write the truth table for the next state function. Your truth table should have one row for every combination of possible input values and state.

$($d$)$ Write the truth table for the output. $($Note that you need at least two bits to represent output$)$

$($e$)$ Using D$-$Flipflops to represent the states implement the FSM$.$