A Devilishly difficult puzzle (I am the author of the puzzle)

Some things that appear not to work

Normally posting answers that merely try things and fail is frowned on around here, but in this case I'm making an exception, since it's been a few days and the question has been bountied -- and also because the things I've tried seem to be very much the sorts of things the question hints at, and it feels like I've tried those particular things quite exhaustively. (Though also fallibly.)

Suppose we

and then

then

The fact that

but, alas,

There are many other similar things we could try. I wrote a little program

I also tried

which also

I wondered about

but

Of course

Some comments on the hints (too bad this little coda spoils the joke at the end of the previous paragraph, but it really doesn't belong anywhere earlier in the answer):

Another thing one could try, I guess, is

Perhaps