What is the general design of these new math solving systems? [D]

From what I've seen online so far, the description of these systems is roughly:

They asked the model (often Aster) to generate statements in LEAN and then submit those to a LEAN compiler to be checked. Based on the results of attempting the LEAN compilation, they somehow add those statements as fact. When the full proof in LEAN compiles, the system is finished.

I can imagine trying to jam as much of a proof as possible into the context window but some of the papers these systems have produced are hundreds of pages. To me this would indicate that somehow the paper is being built piece by piece and being assembled before being submitted to LEAN. This resonates with the part of my understanding that after checking LEAN compilation there's some kind of management of "facts."

I would like to try to implement my own janky version and see if it can answer a question I have about higher dimensional geometry. I'm struggling to find a meaningful way to compose larger ideas from smaller ones. I can imagine it's relatively simple if you know what to do.

What things have you seen? Do you have any ideas you haven't seen that might be interesting to try? Is this a fool's errand because you really need huge amounts of hardware to do anything meaningful? I would welcome any thoughts or links on the matter, cheers