Reasoning involves making accurate inferences based on the information provided in the current context, rather than recalling arbitrary facts from the training data.

Yes, that's what I said. The whole point of hallucinations is that they aren't "arbitrary facts recalled from the training data". They represent attempts to synthesize (i.e., infer) new facts. But because the inferences are not accurate, and because the synthesis process is not sound, the attempt cannot be called reasoning.

It is equally possible to "reason" about things you already know, as about things you've just been told. In fact, the capacity to speculatively, without prompting attempt such reasoning is a big part of cognition.