$2.(25$ pts$)$ Arithmetic ROM Design $($In the notes there is a solved similar problem$)$ Design a digital logic circuit as a Read Only Memory $($ROM$)[$Decoder plus OR gates$].$ A ROM accepts a three$-$bit number and generates an output binary number equal to three times the input number, plus one.

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

$\backslash$hline decimal & x & y & z & $\backslash (\backslash$ldots $\backslash )$ & $\backslash ($ A$\_\{1\}\backslash )$ & $\backslash ($ A$\_\{0\}\backslash )\backslash \backslash$

$\backslash$hline $0$ & $0$ & $0$ & $0$ & & & $\backslash \backslash$

$1$ & $0$ & $0$ & $1$ & & & $\backslash \backslash$

$2$ & $0$ & $1$ & $0$ & & & $\backslash \backslash$

$3$ & $0$ & $1$ & $1$ & & & $\backslash \backslash$

$4$ & $1$ & $0$ & $0$ & & & $\backslash \backslash$

$5$ & $1$ & $0$ & $1$ & & & $\backslash \backslash$

$6$ & $1$ & $1$ & $0$ & & & $\backslash \backslash$

$7$ & $1$ & $1$ & $1$ & & & $\backslash \backslash$

$\backslash$hline

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

$($a$)$ What is the size of the initial $($unsimplified$)$ ROM $?$

$($b$)$ What is the size of the final $($simplified$)$ ROM $?$

$($c$)$ Show in detail the final memory layout.