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Snatch works on the following moves.
Snatch will take the first "snatchable" move used, and not attempt to Snatch any others.
If two Pokémon use Snatch on the same turn, the faster Pokémon will Snatch the first Snatchable move, then the slower Pokémon will Snatch that move from the faster Snatcher, meaning only the slowest Snatcher on the field will get the bonuses from whatever is being Snatched.
If a Pokémon Snatches Psych Up, Psych Up will target whichever Pokémon used Psych Up.