Hey, thanks. This was a fun problem. I got a little bit obsessed though, and I wonder if it might be possible to finish the problem on pencil and paper. I am sure I can finish an approximate version, but I am not sure I can get 12 digits. Did you try something like this?

P.S. I must apologize for abstractly referring to a solution approach as "this", but I didn't want to create a spoiler in case anyone else had felt inspired.