Q: I can not access to mail on iMac.

The story go like this. I forgot my user password and typed a lot of mistakes password. So I change the password with master password. When I logged in, it show that it need keychain password. Keychain password was the perious password but I was able to access it. I don't want to forgot this keychain password so I changed the password. I also changed the Apple ID password because it was also the perious user password. Then, it begin... The facetime, message, mail, and everything I logged in with apple id password. I was able to access the everyting expect mail. I try gmail password, apple id password, user password and every password. It said "This account (Gmail) appears to be offline due to a network problem. Do you want to try to go online now?". I looked at my network and it is completely fine. My grammar might be wrong but please help me.