There is a story over internet and the story goes like this
When the time came for the departure of Rama, Yama says to Rama that he cannot able to come to earth because of the presence of Hanuman. Then Rama throws his ring somewhere and asks Hanuman to bring it back. While searching it, Hanuman realizes that time is cyclic in nature by watching many rings of Rama. In between Rama departures from Earth.
The story seems to be not authentic. But is there any Ramayana that contains a story similar to this?