prove that the unique language $A$ such that $AB$ is context free for all languages B is the empty set

Prove that the unique language $$A\subseteq \Sigma^*$$ such that $$AB$$ is context free for all languages $$\subseteq \Sigma^*$$ is the empty set.

If $$A$$ is not the empty set, there should be a way to construct a language $$B$$ so that $$AB$$ is not context-free. I'm not sure how to generalize this for arbitrary nonempty languages $$A$$ however. The language $$\{0^n 1^n 0^n : n\in\mathbb{N}\}$$ is known to be non-context-free. Also, removing finitely many strings from a non-context-free language results in a non-context free language.

Edit: perhaps modifying the language given will work.

If $$AB$$ is context-free, so should be the set of all its suffixes. Those include the suffixes of $$B$$, and if $$B$$ is complicated enough prefixing $$A$$ does not change the complexity of the set of suffixes.
Your solution $$B = \{0^n1^n0^n \mid \dots \}$$ seems to work, but if $$A$$ ends in $$0^*$$ the reasoning becomes a little complicated. I would suggest $$B' = \{10^n1^n0^n \mid \dots \}$$.