You can read about my personal experience with this effect in this thread including a summary of readings and amount of chlorine in this post. Basically, your CC will take a roughly equal amount of chlorine, or perhaps some less, to get rid of it, but you may also still have ammonia in your pool from the bacterial conversion of CYA so adding chlorine in that case might increase CC even further in the short-run. There's not much you can do except oxidize it all away and chlorine is the least expensive and effective oxidizer for that purpose.