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[Avatar: The Last Airbender] After Azula seemingly kills Aang in Lake Logai, why did Ozai attemp to exterminate the Earth kingdom instead of the Water Tribes?

At the beginning of Book 3: Fire, Aang wakes up on a captured Fire-nation ship, and Sokka excitedly says something like "you're our biggest advantage - the whole world thinks you're dead!"

Even Azula, who made the then-fatal Lightning attack, thinks he is dead, and attempts to glorify Zuko by giving him credit for the attack. Only during the attempted invasion on the Day of Black Sun does Aang reveal himself.

The Avatar cycle seems to be common knowledge, so Ozai knew that the next Avatar would come from a Water Tribe. 

Both the Water Tribes as well as the Earth kingdom are severely weakened by the events of Book 3. Water benders have been systematically removed from society by the Fire Nation, at least in the South. The only reason the kingdom in the North didn't fall was because of Aang's help. And Ba Sing Se, the biggest and most powerful city-state in the Earth kingdom, had just been secretly taken over by the Azula. Other places in the Earth kingdom have varying degrees of Fire Nation control, but Omashu had fallen, Fire colonies dotted the coastline, and about the only retaliation Earth citizens could attempt was isolated pockets of resistance using hit-and-run guerrilla warfare.

Basically, there's no big opponents left that would take significant efforts to conquer on the part of the Fire Nation, and at this point it's only a matter of time until total Fire Nation victory.

So why didn't Ozai immediately dispatch 2 fleets to wipe out the remaining Water citizens? Or, if he was concerned about making another attempt at fighting the Water capitols on their own ground, why did he not attempt to invade the Northern Tribe during Sozin's comet (as opposed to attempting to burn the entire remaining Earth kingdom)? 

Admittedly, the Water Tribes would require over a decade to train the new Avatar, so Ozai wasn't pressed for time. However, attacking the Water tribes to conquer them, and even attempting to capture any newborns (in order to keep the new avatar alive so they would have to hunt for the next version of him / her in the Earth Kingdom) seems like a better tactical choice.
gpt
Didn't Azula strike Aang while he was entering the Avatar state? If you die in the Avatar State, you die in real life. The remaining Fire Sages were on Ozai's side at that point, so he probably knew that, and assumed the Avatar Cycle had been broken.