Mavericks mail will not setup

Question

This is on my familiy's 2013 family iMac. It is running Mavericks. There are five users on the machine, but the one impacted is my daughters.
There is a separate issue where a Pages document created on my daughter's iPad mini will not open with the iMac's Pages app. When this happens, the only way I have ever been able to fix it is to logout of iCloud, lougout of her account on the Mac, and then log back in to iCloud. This used to happen a lot. However, this is the first time in a while, but now her account is in a strange state.
I've logged back into her iCloud account, and everything setup properly except Mail. Mail is checked in the iCloud preferences, but the following is what I observe.
1. When I launch Mail, it prompts me to setup an account. When I enter the account information, Mail does come up, but nothing is there. Mail->Preferences is grayed out, and the application will not Quit with Cmd-Q. If have to force it to quit. After forcing Mail to quit, it will ask me to add the account again if I bring it up.
2. System Preferences -> iCloud will not allow me to uncheck Mail. It complains that I need to go to Mail and disable the account there. Unfortunately, number 1 prevents that.
3. System Preferences -> iCloud will not allow me to sign out of iCloud. It complains that Mail prevents it.
4. System Preferences -> Internet Accounts will not allow me to remove the iCloud account. Mail prevents it again.
This iCloud account works perfectly on her iPad, but now it is toast on the iMac.
Note, please don't respond with lmgtfy. I've been searching for hours, and lots of people seem to be having similar issues to this. However, none of the fixes (where there were any) worked. I've also tried all my usual tricks of removing com.apple.mail, mail data, iCloud folders, etc. Nothing is making her email happy.
I'm guessing this is a keychain issue, but I can't seem to find the right incantation to solve it.
Help!

Answer 1

I had the same issue...I called apple support and they gave me the following instructions. Seems to be working perfectly fine.
- Go to Apple Mail
- Hit on Preferences, go to Accounts and hit the "+" button
- Select "Add other mail account", and hit continue
- Enter your Full Name, email address, and password (DO NOT HIT CREATE)
- Push down the option key, and while holding down the option key hit "NEXT"
- It will route you to the Incoming Mail Server Info window, here you will type in the Mail Server as: imap.gmail.com. Again type in your user name (complete email) and password. And again, hold down the option key, and while holding down the option key hit "NEXT".
- Then it wil route you to the Outgoing Mail Server. Type in the Outgoing Mail Server as: smtp.gmail.com, and again your email and password.
-Hit Create
Once you see the email account on your Apple Mail sidebar, go to compose a new message and send yourself an email to the new email account from your new email account
This worked for me and synced all my folders. Hope it works for you as well.