Can't update apps in iTunes after setting up two step authentication

I set up two step authentication last week, and now, I can't seem to update apps via iTunes. When I click on the availabe update under Apps, and click the Update button, i get prompted for my Apple ID and password (that's normal). However, when I put in my password and click Continue, the dialog box goes away, iTunes says "Accessing iTunes Store" and then the password dialog box pops up again. If I put in my passowrd again, it just keeps doing this. I know that I'm using the correct password, because if I put in the wrong password, the dialog box shows red text that says the password is incorrect. But no matter how many times I put in the correct password, it continues to prompt me, and never downloads the updated app.

Any ideas on what's going on here?