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The Training Cup is a relatively easy cup that allows the player to learn the basics of Flick Rush. The battle is against relatively weak Dream Eaters.
The opponent will open with their Meow Wow, and that shouldn't be hard to beat. The Meow Wow is rather underpowered and doesn't have a lot of attacks to offer that deal massive damage. Any grouped attack should be able to win out over the Meow Wow's attack or defense.
The Hebby Repp is more annoying; it tends to get larger values, and has more devastating attacks. I'd recommend changing to your heavy-hitter when the Hebby Repp comes in, so that you can get this Dream Eater out of there quickly.
Finally, there's the Komory Bat. While it has the tendency to dish out high-level attacks, any fast and/or powerful Dream Eater will easily put it to rest. Despite being a bat, this Komory Bat does not act quickly and can be easily overpowered with some decent attacks.