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Since you was denied entry for the first time, it is safe to apply visa first.
There will be less chance for another refused entry, as your circumstances will be evaluated beforehand. Make sure all the documents are in order.
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Official advice on https://www.gov.uk/check-uk-visa/y/usa/tourism is that you may want to apply for a visa if you’ve previously been denied entry.
If you do attempt visa-free entry, as well as having a return ticket you should bring the same documents you’d need to apply for a visa, to show to officers at the UK border. So, you’d need eg bank statements, proof of ties to the US (employment contract, pay slips), letter from your boyfriend proving accommodation.