The answer is somewhat controversial, because it depends on the interpretation of quantum mechanics, which is not unique. Perhaps the most intuitive interpretation is the Bohmian one, according to which particle is a pointlike object the velocity of which depends on the value of wave function on the position of the particle. Thus, particle takes only one path, depending on the initial position of the particle.
Actually the mirror is not very important, it only plays an auxiliary role by directing the wave beam into a desired direction. A more important peace of apparatus is the beam splitter, which splits the initial wave beam into two beams. In the Bohmian interpretation, the wave is splitted but the particle is not.