Question $2$ Pushdown Automata and Context$-$Free Languages$\}$ Consider the alphabet $\backslash (\backslash$Sigma $=\backslash \{$a$,$ b$,\backslash$#$\backslash \}\backslash )$ and the language $\backslash [$ B $=\backslash \{$ w $\backslash$in $\backslash$Sigma$^*\backslash$mid w $=$ a$^$n $\backslash$# b$^\{2$n$\}\backslash$# a$^$n $\backslash$text$\{$ for some $\}$ n $\backslash$in $\backslash$mathbb$\{$N$\}\backslash \}.\backslash ]\backslash$begin$\{$enumerate$\}\backslash$item$[($a$)]$ Prove that $\backslash ($ B $\backslash )$ is not context$-$free. $\backslash$item$[($b$)]$ Explain how the following PDA $\backslash ($ P $\backslash )$ operates on words over $\backslash (\backslash$Sigma $\backslash )$ and show that all words in $\backslash ($ B $\backslash )$ are accepted by this PDA. The PDA $\backslash ($ P $\backslash )$ is described with the following transitions: $\backslash$begin$\{$itemize$\}\backslash$item $\backslash ($ q$\_0\backslash$xrightarrow$\{\backslash$epsilon$,\backslash$epsilon $\backslash$to $\backslash$$$\}$ q$\_1\backslash )\backslash$item $\backslash ($ q$\_1\backslash$xrightarrow$\{$a$,\backslash$epsilon $\backslash$to a$\}$ q$\_1\backslash )\backslash$item $\backslash ($ q$\_1\backslash$xrightarrow$\{\backslash$#$,\backslash$epsilon $\backslash$to $\backslash$epsilon$\}$ q$\_2\backslash )\backslash$item $\backslash ($ q$\_2\backslash$xrightarrow$\{$b$,$ a $\backslash$to $\backslash$epsilon$\}$ q$\_2\backslash )\backslash$item $\backslash ($ q$\_2\backslash$xrightarrow$\{\backslash$#$,\backslash$epsilon $\backslash$to $\backslash$epsilon$\}$ q$\_3\backslash )\backslash$item $\backslash ($ q$\_3\backslash$xrightarrow$\{$a$,\backslash$epsilon $\backslash$to b$\}$ q$\_3\backslash )\backslash$item $\backslash ($ q$\_3\backslash$xrightarrow$\{\backslash$epsilon$,\backslash$$ $\backslash$to $\backslash$epsilon$\}$ q$\_4\backslash )\backslash$item $\backslash ($ q$\_2\backslash$xrightarrow$\{$b$,\backslash$$ $\backslash$to $\backslash$$$\}$ q$\_5\backslash )\backslash$item $\backslash ($ q$\_2\backslash$xrightarrow$\{$b$,$ b $\backslash$to b$\}$ q$\_5\backslash )\backslash$item $\backslash ($ q$\_5\backslash$xrightarrow$\{\backslash$epsilon$,\backslash$epsilon $\backslash$to b$\}$ q$\_2\backslash )\backslash$end$\{$itemize$\}\backslash$item$[($c$)]$ Identify the language $\backslash ($ L$($P$)\backslash )$ of the above pushdown automaton. $\backslash$item$[($d$)]$ Provide a context$-$free grammar for the language $\backslash ($ L$($P$)\backslash ).\backslash$end$\{$enumerate$\}$