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Learned an interesting thing through browsing posts and clips.
In the "Berry Catch" minigame, Eevee has to catch berries while avoiding symbols of fire, water, a leaf, lightning, and a snowflake. If Eevee catches one of those, losing the game, the one it caught will linger on the screen afterward, with Eevee staring at it confused until another button is pressed.
As no one has evolved their Eevee yet, we don't know if it definitely influences evolution, but if it does, then we can guess how it might influence the evolution of Flareon, Jolteon, Vaporeon, Leafeon, and Glaceon. That would leave Espeon, Umbreon, Sylveon, and the three "secret" forms unaccounted for.