Can you successfully do a PRAM reset?
When you do that reset, continue to hold the same 4 keys (Option-Command-P-R) until you hear the boot chime two more times, then (while the chime is still sounding), release the keys, then immediately hold Command-R
If your Logitech keyboard is a Windows keyboard (with a windows key, not a command key), make sure that you hold Windows - R keys, not alt - R.
If that doesn't work, try to boot to Internet Recovery. Same method, but hold Option(alt)-Command-R during boot.
If you see the Apple icon, rather than the rotating globe during boot, then that still doesn't work.
(related question - Have you EVER booted successfully to your recovery system with this keyboard?)
Try a different keyboard - an Apple-brand one, if you have access to one.