$(20$ marks$)$ Given the following grammar:

S $->$ aS $|$ bS $|$ cSa $|$aAaB $|$AbBbA

A $->$ baB $|$ bC $|$ a $|$ c

B $->$ abA $|$ bC $|$AC $|$a

C $->$ aA $|$ baB $|$ bbb $|$ a

$($a$)(10$ marks$)$ Construct the PDA that accepts the same language by empty stack.

$($b$)(10$ marks$)$ Construct an equivalent PDA that accepts the same language by final states.