Consider the Pushdown Automaton $($PDA$)$ for the language L$=\{$an$+1$b$2$n $|$ n$>0\}.$

The following relate to the PDA:

$\backslash$Sigma $=\{$a$,$b$\}$

$\backslash$Gamma $=\{$a$,$Z$\}$

Q $=\{$q$0,$ q$1,$ q$2,$ and q$3\},$ where q$0$ is the initial state and q$3$ is the final state.

Negative grading applies.

Do NOT include any spaces between characters, symbols, or punctuation marks. Write everything as one string, excluding spaces.

If you want to include something like q$0,$ type it out as q$0.$ Do not paste anything except for special symbols such as $\backslash$epsi $,\backslash$delta $,$ etc.

Take Z as the bottom$-$of$-$stack indicator and assume that it is already in the stack

$1.)$ Complete the transition function for the PDA

You may not add or change the name or order of states.

$\backslash$delta $($q$0,$a$,$Z$)=($Answer $1$ Question $11$

$)$

$\backslash$delta $($Answer $2$ Question $11$

$)=($q$1,$aa$)$

$\backslash$delta $($Answer $3$ Question $11$

$)=($Answer $4$ Question $11$

$)$

$\backslash$delta $($q$1,$b$,$Answer $5$ Question $11$

$)=($q$2,$Answer $6$ Question $11$

$)$

$\backslash$delta $($q$2,$Answer $7$ Question $11$

$)=($Answer $8$ Question $11$

$)$

$\backslash$delta $($q$2,$Answer $9$ Question $11$

$)=($q$3,$Z$)$

$2.)$ Using the correct PDA from $(1)$ above, provide the turnstile notation using the string "aabb".

$($Answer $10$ Question $11$

$)($Answer $11$ Question $11$

$)($Answer $12$ Question $11$

$)($Answer $13$ Question $11$

$)($Answer $14$ Question $11$

$)($Answer $15$ Question $11$

$)$