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$10k sounds high, even for a JCW. The 2012 has an N14 engine, and the price should reflect that. Expect having to perform a timing chain service, at the least.
I have a 2012 R58 equipped with the N18
Look at the sticker under the hood.
Passenger side circled in blue
2012 was a transition year. The Cooper S models shipped with the N18 engine, but the JCW models were still being built with the N14 engine. Your Coupe looks to be a Cooper S model.