You could do it with more than two grooves, just to having them at 360/n degrees apart. You’ll just have to make the groove spacing wider as the number of tracks go off. Of course that comes at the cost of playback tine.
EDIT- Apparently it was the K-Tel Superstar game
https://boardgamegeek.com/boardgame/4521/k-tel-superstar-gam...
https://www.discogs.com/master/19554-De-La-Soul-Me-Myself-An...
Side 4 has a double groove, which would give you either The Great Escape + Made again (a sort of a happy ending) or The Great Escape + 20 minutes of water sounds (which can be interpreted as the sad ending).
Reference: https://en.wikipedia.org/wiki/Brave_(Marillion_album)#Vinyl_... (I also have a copy and can confirm indeed it works like that).