Consider the language $L=\{\begin{array}{ccc}\text{w}\text{i}\text{n}& \{a,& b{\}}^{*}\text{:}w\end{array}$ contains exactly $2$ as or exactly $\{$:$3b(s)\}.$

$[25$ marks$]$

For example, $\{aabbbb,$bbbababbb,$aa,bbb,$baaabaaaba,babab$\}subL$ but ababa!inL.

$($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$]$