$(5$pts$)$ Design a context$-$free grammar $($CFG$)$ that generates all strings over $\{a,b,c\}$ that contain at least one $\text{'}$a$\text{'},$ one $\text{'}$b$\text{'},$ and one $\text{'}$c$\text{'}.$

I. Define the rules of the CFG$.$

II$.$ Provide a derivation for the string "abc".

III. Explain why the CFG does not generate the string $"$ab$"$ or $"$bc$".$