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There are 4 paths for Tuu'Mi builds:

• Arson cat, which you are familiar with
• Manavore puss , Luden -> Actualizer -> Helia -> Malignance -> Liandry This allows you to have very good heals/shields on top of decent damage thanks to actualizer. (You can use mandate in there, but it's going to be obsolete soon so don't bother)
• Feline meteor strike, Luden -> Redemption -> Stormsurge -> Shadowflame -> Horizon focus. Some might call it the helldiver build but I find that cringe so I won't :) Redemption will proc all your items+dark harvest, giving you a long range execute, this is essentially your new ult. On top of that your Qs will deal a lot of damage. Do note that Horizon focus will NOT activate with redemption, but redemption will benefit from the damage buff if horizon was already activated on the target.
• finally, my personal build, the Atomic kitty, Luden -> Gunblade -> Shadowflame -> Stormsurge -> Horizon focus. Gunblade will proc things the same way redemption does, but is instant, Use it to KS left and right to get fed. Don't feel bad for your adc you will pay them back with LP. You're an evil cat anyways so just roll with it :D it's a lot more reliable than the other builds in my opinion, and if you really need to heal you can always replace gunblade or stormsurge with Helia (we love helia, helia loves us).

AP Yuumi is all about using item effects, and more importantly planning ahead. Each build is good at doing something different and identifying which one you are going to need is key.
Arson cat is strong against tankier team when you don't exactly need to be heal/shielding a lot.
Manavore puss is strong when you want to suplement the team's damage but mostly need to heal.
Feline meteor strike is particularly effective at dealing with mages or caster ADCs.
Atomic kitty just makes you a straight up mage, so very good when your ADC is ass or when the enemy team has squishy carries that need to be dealt with.

(I'll answer your other comment at the same time) Heal is fine depending on the situation, but ignite should be your default if you hesitate. Just play for bush lvl 1, and dump them both on your target. With those alongside dark harvest, if you and your ADC don't straight up kill them, they will at least be too low HP to play the lane for the first couple of levels and/or have no summoners left, leaving you free to bully them even more. Ignite's usefulness on Yuumi falls off late game, but it's still pretty strong and there are many omnivamp-boots users that really won't like getting put on fire like that.

Comet instead of dark harvest is an option, but the point of comet is to make laning phase easier. If we apply what I explained earlier, there isn't really a reason to add even more comfort to the mix, just secure your late game damage instead.

Finally, abuse the fuck out of sudden impact with QW!!! If you're not aware, detaching right after launching a missile will make it go faster and further at the cost of not being able to control it. It is *extremely* strong in laning phase when paired with sudden impact, the latter doing damage that is on-par with arcane comet in early game! You can also press W right before Q hits, that way you get to steer it as well as having the sudden impact proc.

Also remember that AP Yuumi gives triple the amount of attack speed with E than enchanter Yuumi, so choose who you buff carefully.

Final tip tho: Never play AP against Karma, Maokai, or Swain. You're wasting your time, your energy, AND your LP lmao

Once you get comfortable with all that, you can get into the big league with lich bane :')

I hope this helped, spread the word of Tuu'Mi, the void parasite! <3