You are correct. The amenity fee gives you access to outdoor activities and inside recreation centers. However, the fitness clubs (the name for the workout facilities at some of the recreation centers) require a separate fee for their use. Many recreation centers have outdoor exercise machines that you can use without extra charge. They use your bodyweight to get you sweating. You could try them before giving a fitness center a try. Instructions are printed on the machines, but you can also ask recreation center staff how to use them.
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