Comment by GeorgeDewar

That writing isn't the only "thinking" though. Some thinking can happen in the course of generating a single token, as shown by the ability to answer a question without any intermediate reasoning tokens. But as we've all learnt this is a less powerful and more error-prone mode of thinking.

So that is to say I think a small amount of secret reasoning would be possible, e.g. if the location is known or guessed from the beginning by another means and the reasoning steps are made up to justify the conclusion.

The more clearly sound the reasoning steps are, the less plausible that scenario is.