What musical liturgical melodies could Jesus have sung in the synagogues of his day?

What musical liturgical melodies could Jesus have sung in the synagogues of his day? Do any possible examples still exist, based on historical evidence? Have any musically inclined historians mentioned any possibility of this and have given examples of what Jesus and his Disciples sang or chanted liturgically in their synagogues of the first century AD? Bonus question: Are there any melodies that exactly or nearly are the same as found in early Gregorian Chant or Early Byzantine Chant?