Please answer the question if the answer is based on math please don't focus in the explanation just solve the math correctly with an structural way..Thank you.Let $M$ be the pushdown automata $($PDA$)$ defined as:

$M=(\{q_0,q_1,q_2\},\{a,b,c\},\{A\},,q_0,\{q_2\})$

$$:$,(q_0,a,)=\{[q_0,A]\}$;$,(q_1,,)=\{[q_2,]\}$;

$,(q_0,b,A)=\{\begin{array}{cc}[q_1,& ]\text{;}\end{array},(q_2,c,)=\{[q_2,]\}.$

$,(q_1,b,A)=\{[q_1,]\}$;$,$

a$)$ Give the state diagram for the PDA $M_1.$

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