<p>I agree that energybending is not a prerequisite to the Avatar State and vice-versa. A small point though: before Aang bends Ozai's energy, Aang has enough control to go out of the Avatar State out of his own will to stop the past Avatars from killing Ozai. In terms of Aang's ability to go in and out of the Avatar State, I think it's fair to say that he has figured out at least that. Shortly after he energybends, Aang shows control of the Avatar State to put out the fires.
</p><p>However, I don't think it's a coincidence that both Aang and Korra first show their ability to energybend very near the time they show control of the Avatar State. When they open their seventh chakra, they see a vision of their higher selves. By going there, they are able to control themselves completely in the Avatar State. Specifically for Korra, she can energybend herself only when she connects to her inner spirit via the Tree of Time. I connect this to the Lion's Turtle's talk of needing a unbendable spirit to bend another's energy. The ability to have complete control of your own thoughts and actions can apply to both controlling yourself in the Avatar State and bending your own energy. Although the Avatar State is not required for energybending, I think they both involve similar spiritual revelations.
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