# Is $P^{SAT[1]}=NP \cup coNP$

I have a following problem:

Let $$P^{SAT[1]}$$ be a class of problems decidable by a deterministic polynomial Turing Machines with SAT oracle. (only one question to oracle).

Assume that: $$coNP \neq NP \neq P$$

Decide if $$coNP \cup NP \neq P^{SAT[1]}$$ or $$coNP \cup NP = P^{SAT[1]}$$.

I know how to show that $$coNP \cup NP \subseteq P^{SAT[1]}$$.

But I have no idea what about inverse direction. Any tips?

Hint (noting the homework tag on the other post): $P^{SAT}$, by the Cook-Levin theorem, is the same as an $NP$ oracle. Given that $\text{co-}NP \neq NP$, what does this imply?