The system is right. If you pass, I can choose to pass, ending the game, or to play(if I think there is something that I can do to get more points).

Sometimes, if you play, you will lose points(playing a stone where there is no way for it to live, thus giving a free point for your opponent) and then you should pass. Otherwhise, you should try to get more points.

If you pass, and your opponent does not, you can pass again. If that was an overplay(a move that gives nothing to the person who plays it, or a move that are bad for them) you will gain one point. Just be careful because overplays, if not answered, sometimes can cause damage.