Consider the follow

ing CFG$,$ which we call G$1$

A $->0$A$1$

A $->$ B

B $->$ #

$($a$)$ Prove that $000$#$111$ is a derivation of G$1$

$($b$)$ Draw the state diagram of the pushdown automata corresponding to G$1.$ Call such PDA PDA$1.$

$($c$)$ Show step$-$by$-$step that the string: $00$#$11$ is accepted by PDA$1.$