Continue with your normal circuit keeping agood lookout and avoiding collisions.
(a) If the other aircraft turns final in front, lands and vactes in time for you to land then no problem.
(b) If the other aircraft turns final in front and causes you to go arround then they are in the wrong.
If your circuits are of the correct size i.e. not too big then the only possible outcome from what you describe in general should be (b)
unless the other aircraft follows you round.
Regards,
DFC