[masfuerte]

I suspect it's not possible (as an end user) to get a thinking trace from one of the models. But what happens with "thinking" is that the model has a conversation with itself in an attempt to home in on a better answer to the original prompt.

The "amount of thinking" is how long this internal conversation is allowed to progress. The longer it goes on the more it costs. It's all part of the token budget but, because this internal dialogue is hidden, it's not obvious to the end user.