Let $M$ be the PDA defined as: $M=(\{q_0,q_1,q_2\},\{a,b,c\},\{A\},,q_0,\{q_2\}),$ :

a$)$ Give the state diagram for the PDA, M$.$

b$)$ Trace all computations of strings $ac,$abc,aabcc 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$).$