Please answer the question, if it is math just solve the math correctly no need to pass the time for explanation please.Thank you.Let $$be the pushdown automata $($PDA$)$defined as:

$=(\{0,1,2\},\{,,\},\{\},,0,\{2\})$

$$:$,(0,,)=\{[0,]\}$;$,(1,,)=\{[2,]\}$;

$,(0,,)=\{[1,]$;$,(2,,)=\{[2,]\}.$

$,(1,,)=\{[1,]\}$;$,$

a$)$Give the state diagram for the PDA $1.$

b$)$Trace all computations of strings aabbc,abbc,$$in M$.$

c$)$Give a set$-$theoretic definition of the language accepted by M$.$

d$)$Write a grammar for the corresponding PDA.

e$)$Write the shortest string that is IN and NOT IN the language L$($M$).$