  We can only perform a finite rounds of measurements in protocols with local
operations and classical communication (LOCC). In this paper, we propose a set
of product states, which require infinite rounds of measurements in order to
distinguish this given set of states perfectly by LOCC. Therefore, we can
conclude that these sets of states are locally indistinguishable. More
accurately, given any multipartite LOCC indistinguishable set where every local
system cannot start with a nontrivial measurement, then after appending these
states with arbitrarily choosing two nonorthogonal states, we obtain another
LOCC indistinguishable set. It can be seen that some parties can perform some
nontrivial measurements. Hence, these sets are quite different from those
constructed before. This result broadens the knowledge of nonlocality without
entanglement to a certain extent.
