What exactly happens when Ты are reborn?

In real life, I don't believe in rebirth. But in the Percy Jackson universe, if Ты choose to be reborn, Ты will first be immersed in the river Lethe in the underworld, which will leave Ты with completely no memories, then the normal process continues.

i think, Ты live life again as a different person, but Ты still have your personality. Ты need to be very kind cause Ты could probably be sent to the Fields Of Punishment.

Ты become alive again as a new person except with the same soul and personality. Then Ты get another chance to achieve Elysium. If Ты acieve Elysium three times Ты get into the Isles of the Blest.