There’s a very simple way to root Verizon's S3 with 4.3 with saferoot. Here's the link. I know it says that it’s for the S4, but it works perfect with Verizon’s S3 also. Like GSDer said though, once you update to 4.3 you won’t be able to go back and DON’T try to install a rom.
Also, for...