Wow, what has happened to your drive?
So, there's a strange issue going on with your partition, and you will need to back up your drive to an external drive, then erase the internal. Reinstall your OS X using your internet recovery, then restore files from your external.
Did you already talk to Apple about this? Then, that's very much what they told you, and that's likely your best way to proceed.
It's not a particularly difficult thing to do, but it will take you several hours. There's lots of waiting
(I suspect that you might find out that when you try to erase, that may not work, either. If you get some kind of errors when you try to erase, you will likely need to replace the drive... )
You MIGHT be able to run some disk repair utility, such as Disk Warrior, which might sort things out.
But, you also have to purchase that (unless you already have that software), and a little time with backups, etc, should sort things out.