> Reasoning properly is at least operating through processes that output correct results.

Human "reasoning" (ie speech or self-talk that sounds a bit like reasoning) often outputs correct results. Does "often" fit the definition?

> Which problems that demand producing correct solutions could be solved by a processor which could not solve a "detective game"?

For example, "what colour is the sky right now?". A lot of people could solve this (even if they haven't looked outside), and so could a lot of language models, which can't solve this detective game.