$4.$ Luke needs your help one more time. Luke wants to automate a transport vehicle between $4$ planets: Mercury, Venus, Earth and Mars. Luke has handed you the following notes:

a$.$ I have a remote control that has three buttons: UP$,$ DOWN, RESET. I can only press one button at a time. You may assume that when I press button, let's say UP$,$ the value of UP shall be a$"1".$

b$.$ When I press UP$,$ I want the vehicle to move from Mercury $\backslash (\backslash$rightarrow $\backslash )$ Venus $\backslash (\backslash$rightarrow $\backslash )$ Earth $\backslash (\backslash$rightarrow $\backslash )$ Mars $\backslash (\backslash$rightarrow $\backslash )$ Mercury and the whole cycle repeats itself

For example: If the vehicle is on planet Mercury, pressing an UP button will move the vehicle to Venus instantaneously.

If the vehicle is on planet Mars, pressing an UP button will move the vehicle to Mercury instantaneously.

c$.$ When I press Down, I want the vehicle to move from Mars $\backslash (\backslash$rightarrow $\backslash )$ Earth $\backslash (\backslash$rightarrow $\backslash )$ Venus $\backslash (\backslash$rightarrow $\backslash )$ Mercury $\backslash (\backslash$rightarrow $\backslash )$ Mars and the whole cycle repeats itself

For example: If the vehicle is on planet Mars, pressing a DOWN button will move the vehicle to Earth instantaneously.

If the vehicle is on planet Venus, pressing a DOWN button will move the vehicle to Mercury instantaneously.

d$.$ Drop all passengers off on planet Earth.

e$.$ When I press RESET, I want the vehicle to move to Mercury instantaneously

$4$a$.(10$ points$)$ Draw the State Machine based on the information given above.

$4$b$.(10$ points$)$ Create the State Transition Table based on your State Machine.

Your State Transition Table needs to look like: $4$c$.(10$ points$)$ Create the Encoded State Transition Table based on your State Transition Table. Your Encoded State Transition Table needs to look like:

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{2\}\{|$c$|\}\{$ Present State $\}$ & $\backslash$multicolumn$\{2\}\{$c$|\}\{$ Inputs $\}$ & $\backslash$multicolumn$\{2\}\{$c$|\}\{$ Next State $\}$ & Output $\backslash \backslash$

$\backslash$hline Q$1$ & Q$0$ & UP & DOWN & D$1$ & D$0$ & Drop $\backslash \backslash$

$\backslash$hline & & & & & & $\backslash \backslash$

& & & & & & $\backslash \backslash$

& & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$4$d$.(10$ points$)$ Find the Boolean Expression for D$1,$ D$0$ and Drop using either K$-$map or Boolean Algebra. You must show your work to get all the credits.

$4$e$.(10$ points$)$ Draw the final circuit. Do not forget to add the D flip$-$flops.