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Your score seems fine for recieving a sizeable CLI from Synch, but I would wait until the first statement cuts and you PIF. No need to rush for a credit limit increase with Synchrony, they will practically throw them at you. Show some usage and payment activity and then bombs away.
Call the back door number 877-379-8173. Ask for 10K or 15k if you feel lucky. It doesn't matter which one. They will counter with their max offer.