Design a PDA to accept each of the following languages. You may design your PDA to accept

either by final state or empty stack, whichever is more convenient. Give your solution as a

transition diagram, using the notation of the Hopcroft$-$Motwani$-$Ullman textbook.

$($a$)\{\begin{array}{c}{a}^{n}x\text{:}n0\\ \text{x}\text{i}\text{n}& \{a,& b{\}}^{**}\end{array}$ and $\{$:$|x|n\}.$

$($b$)\{\begin{array}{ccc}\text{w}\text{i}\text{n}& \{a,& b{\}}^{**}\text{:}w{?}^R\end{array}\{$:xin$\{a,b{\}}^{**}\}$