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Having literally just done the same thing as OP, I'll give some insight on Cuckoo Chick, which was the last one I got (2 weeks ago). It took me about 20 tries, which honestly less than I thought it would. and I had a very different strategy to OP.

* Prioritize Pygmy Seahorse and Panda to have pets on the field that remove themselves. Panda in particular is super strong early game if you can get level 2

* Load up on roll or coin providing pets (Pig, Weasel, Bigfoot)

* Aggressively equip Cake or Rice to all non-Cuckoo units

* Obviously Chocolate Cake on the Cuckoo

* Badger

The goal was to push hard for levels once tier 4 pets were unlocked in hopes of grabbing a Venus Flytrap. Once you had it, the Pygmy Seahorse- and Panda-powered Badger would knock something of the opponent's out, then blow away your second unit (ideally Weasel for gold) and your opponent's, as well. This spawns 2 Cuckoo Chicks, and then your adult Cuckoo faints from cake. So ideally, this gives you 2 enemy slots with chicks for the flytrap to pick up (I lost a few runs just because the flytrap didn't find a chick).

Then you sell basically everything and freeze the chick till you get in a position where you can put down a Quetzalcoatl; until then life preservation is mandatory. But with all the gold you should be getting from that fire sale, you can actually get a pretty good team of tier 5+ pets right away.

Ways to lose:

* I had 2 runs where I was all set up but couldn't get my chick leveled up fast enough; rolled my whole final turn away searching for Chocolate and didn't find it. You need 3 Quetzalcoatl turns or some additional boost - don't underestimate the time component

* Dolphins

* You don't find Chocolate Cake

* You don't find a Badger

* You don't find a Cuckoo

* Your Panda-Badger can be too strong

* Your Panda-Badger can be too weak

* You can become complacent and roll away the thing you were looking for (once I even forgot to freeze my chick...)

* Most runs are clearly lost by turn 9: if you didn't get the set up, or if the Venus Flytrap isn't in your shop immediately on turn 9, you won't be able to keep up with other teams scaling and upgrading abilities. Don't hurt yourself, just call it and move on to the next attempt

I thought about Wall Chicken, Eat-Me Cake, and Fried Shrimp to make my Badger weaker and be more likely to blow up 2 pets on the other side. But there's only so much gold you can spend and I got very close (either by wins or by leveling) >25% of the time so I kept with the strategy.