Demonstratably, you don't actually experience doing so. When a solution to a problem you were thinking aloud about yesterday suddenly pops up in you mind today, you have 0 idea how your brain came up with it.

What you experience is only the pop ups.

Yes, I have had that experience. And how do we know it just "popped up"? Because we can watch our own consciousness, and we can see that it did not originate at the level of conscious thought.

So you mean unless you think an idea in words, you are not doing anything conscious? Does that mean I play a real-time video game entirely unconsciously? Because I don't think "I need to go there and do this" in words, I just do it.

Seen from my perspective this doesn't make you any more different than the chinese room, as you cannot prove your claim to observe your very own mind while thinking.