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It has been explained that the Avatar cycle can be broken by killing the Avatar while they are in the Avatar State.

Is there another way?

In their hunt for the Avatar, the Fire Nation army killed every Air Bender and Air Nomad they could find. What if the Fire Nation had killed all of the Air Bender/Nomad before Aang's birth - or if they had killed all of the Water Benders first?

Would the Avatar spirit skip to the next element or somehow inhabit someone with a distant bending gene from generations ago - or would the whole cycle just completely break?