Implement the following functions using a single $163$ROM. Use dot notation to indicate the ROM contents.

$x=AB+BC{C}^{\text{'}}D+A^{\text{'}}{B}^{\text{'}}$

$Y=AB+BD$

$Z=A+B+C+D$

Implement the functions from previous problem $($in question $1)$ using a PLA. You may use dot notation.

Provide a design of the sequential circuit for the vending machine controller: One product sells for $$0\mathrm{.}40,$ and the machine takes only quarters $($input $Q)$ and dimes $($input D$)$ with an output signal I for product release and a few necessary output signals for change release $($if there is change to return$),$ for example, $C5$ for a $5-$cent change, $C10$ for a $10-$cent change, etc.. No other signals are used.

$($a$)$ Show the complete state diagram and state transition table.

$($b$)$ How many flip$-$flops are needed? Draw a block diagram of the circuit showing the key blocks, signals and wires.

$($c$)$ How many K$-$maps are needed to derive the complete circuit?

Note: You do not need to implement the FSM using logic gates and flip$-$flops.