What would you usually do? If I were you I'd continue what you do after catching any Pokemon
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No you can't catch Mewtwo in Pokemon silver, you must trade it from Pokemon yellow.
how do you catch a mewtwo on Pokemon platinum
No, Mewtwo cannot be caught in Pokemon Crystal. It can, however, be traded from Pokemon Red, Blue, or Yellow.
Catch 'em all
There is no such Pokemon as "Shadow" mewtwo. There is a regular mewtwo that you can catch like any other Pokemon. It's located in the Cerulean Caves.