# Does both the language and its complement being context free imply it being deterministic context-free?

Suppose $$L \subset A^*$$ is context free and $$A^*\setminus L$$ is also context free. Does it mean, that $$L$$ is deterministic context free?

If it is not, I would like to see a counterexample (I failed to construct one myself).

Note that the converse is true. Moreover, a complement to a deterministic context free language is also deterministic context free as one can simply change labels on the corresponding deterministic pushdown automaton.