Simplify the following grammar:

$S$AaB$|C$

$AaA|$

$BbB|$

$CcC|D$

$DdD$

$($a$)$ Show the process to remove all $-$productions.

$($b$)$ Show the process to remove all unit$-$productions.

$($c$)$ Show the process to remove all useless productions.