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Does the sink drain normally otherwise?
Ok I hate to be the one to say it but you will need to have someone run a power snake into the plumbing at the clean out or just past the "P" trap.
If you like you can remove the "P" trap under the sink. The clog may be there. You say that you have tried drain cleaners to no avail. This tells me that it isnt at the "P" trap but further down the line. Drain cleaners effectiver range is about the "P" trap. Anything past that will require disassembly and a plumbers power snake.
Please post any other concerns here.
No. Not at all. Its the guy that runs it that is.
Please be aware that you can rent a power snake. Please do not do this unless you know how to run one. Getting a power snake stuck in your plumbing makes "pricey" look real cheap!
Best suggestion is to call a plumber or a good handyman.