Consider the context$-$free language L$,$ where L consists of all strings over the alphabet $\{$a$,$ b$,$ c$\}$

such that every $\text{'}$a$\text{'}$ is immediately followed by a $\text{'}$b$\text{'},$ and every $\text{'}$c$\text{'}$ is immediately preceded by a

$\text{'}$b$\text{'}.$ A few examples of strings in this language are $\text{'}$ba$\text{'},\text{'}$bcb$\text{'},$ 'bab', $\text{'}$bbcb$\text{'},$ and $\text{'}$bbbb$\text{'}.$

Construct an informal proof demonstrating that this is indeed a context$-$free language. As part

of your proof, design and present a pushdown automaton $($PDA$)$ that accepts this language.

Make sure your response includes the following:

$1.$ A clear definition of the language L in terms of a context$-$free grammar $($CFG$).$

$2.$ A description of your pushdown automaton, including its states, transition function, and

acceptance condition. You should also provide a diagram of your PDA.

$3.$ A brief informal proof $($in words$)$ explaining why your PDA correctly recognizes the

language L$,$ and thus why L is a context$-$free language.