"Prove that you voted for Putin or you are out of a job".
About the best I can come up with is a QR code displayed on the screen and on a printout that you can compare with a third party phone app. Machine results are tabulated, and the QR code sheet is put in a lock box separately. This at least provides some way to compare what the computer says you voted versus the QR backup ballot for audits. I'm sure there are holes in my idea.
That's definitely not secret. If you can audit it on your phone, baddies can force you to show your phone to verify that you voted "correctly".
And what happens if baddies come to your house before the election, and say that after election day they'll check up on you, and if you don't they'll beat you up?