$(25$ 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.

$($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.