iPhone 5 iOS 7 Beta 6 won't update to public release

My iPhone 5 is on iOS 7 Beta 6 (I didn't update to the GM because they didn't have the OTA update available) and when I tried updating it to the public release it says that it's on the latest version. I thought maybe I just had to do it in iTunes on my computer, but that said the same thing. Tried restarting, that didn't work, either. Any suggestions? I'd prefer not to have to restore, because I just don't always trust the iTunes backups, I've had several problems with them. Do I need to be on GM in order to update to the public release? Or if you're already on the GM do you even have to update again?