$1.(4$ points$)$ Consider the following grammar:

S $->$ AbB $|$ bAc

A $->$ Ab $|$ aBB

B $->$ Ac $|$ cBb $|$ c

Show how the following string aAcccbbc can be generated by the above grammar.

$2.(5$ points$)$ Construct a PDA that accepts L $=\{$anbmcn $|$ n$>=1,$ m$>=1\}$ for aabbbcc

$3.(6$ points$)$ Using the concept of Pumping Lemma, show that $\{$an bn cn $|$ n $>=0\}$ is not context$-$free using two cases:

a$.$ The v part of the string S $=$uvxyz contains only several $$a$,$

and the y part contains several $$c$.$

b$.$ The v part of the string S $=$uvxyz contains several $$a$$ and $$b$$

and the y part contains several $$b$$ and $$c$$