This $is$ the third time I have posted this question. Please, Please, PLEASE READ THE QUESTION. $Do$ not assume the alphabet yourself. The alphabet $is$ right there listed below $\{g,o,l,e\}.$ And the constraints $of$ the Language $is$ listed right there. Once again PLEASE READ THE QUESTION. $DO$ NOT RUSH THE ANSWER. Please $do$ not copy $it$ elsewhere. Provide a detailed answer $if$ you can. And Thank you for your patience with $me.$

Let $L=\{\begin{array}{c}{g}^m{0}^h{g}^k{l}^n{e}^f\end{array}$ where $n,h0,f1,f=m,$ and $2n=k$

a$.$ Construct a final$-$state PDA with that recognizes the strings of L$.(15$ points$)$

b$.$ Convert the generated final$-$state PDA into an empty$-$stack PDA using the

approach studied in lectures. $(15$ points$)$

c$.$ Does the word goggle belong to L$?$ Justify your answer with a sequence of

moves of the empty$-$stack PDA and explain its indication. $(10$ points$)$