$2.[$FSM Optimization, $40$pts$]$ Consider a vending machine that only accepts Dime $(\backslash$$$10)$ and Nickel $(\backslash$$$5).$ When the total input reaches $\backslash (\backslash$$ $15\backslash ),$ the machine dispenses a drink. Once the vending machine is in the "open" state $($drink dispensed$),$ it remains in the open state until a RESET signal is received. The FSM below represents all possible input sequences. Referencing this FSM$,$ answer the following questions:

$($a$)$ Minimize the FSM using a Moore machine. $($Ensure the FSM accounts for all possible inputs in each state, including N$,$ D$,$ RESET, and N$\text{'}$D$\text{'}.)[20$pts$]$

$($b$)$ Encode the states of the Moore machine such that the current state's accumulated amount is represented in increasing order. Use D flip$-$flops, $2-$input AND gates, and OR gates to draw the circuit. $($The initial state with $\backslash (\backslash$$ $0\backslash )$ should be encoded as $\backslash (\backslash$mathbf$\{00\}\backslash ),$ followed by $\backslash (\backslash$mathbf$\{01\}\backslash )$ for the next smallest amount, and so on$.)[20$pts$]$