onsider the language L $=\{$w in $\{$a$,$ b$\}$

$$

: w contains exactly $2$ as or exactly $3$ bs$\}.[25$ marks$]$

For example, $\{$aabbbb$,$ bbbababbb, aa$,$ bbb$,$ baaabaaaba, babab$\}$ L but ababa $$ in L$.$

$($a$)$ Construct a minimal DFA that accepts L$.[6$ marks$]$

$($b$)$ Use Myhill$-$Nerode Theorem to prove that your DFA is minimal. $[7$ marks$]$

$($Hint: derivatives of regular languages might simplify the proof.$)$

$($c$)$ Construct an NFA that accepts L$.[6$ marks$]$

$($d$)$ Construct a CFG that generates L$.[6$ marks$]$