There is sorto of an easier way round this.
There are 3 reasons you can't sign in to gmail.
1. you are in another country trying to access the mail with same credentials.
I have had it where I have signed in and in the same country and it notifies you.
I was trying to help my mum while away and signed into her system, this triggered the security system.
I needed to get a varification from her phone.
After that I had to change her password to one then change it again when she came back.
2. you got hacked.
Once I got someone in my account and it triggered the security system.
However the system grabbed his ip, and other info and this included his isp.
I was able to contact his isp with the information and the issue was handled.
The last time is simply bad luck.
close firefox, and then with ccleaner just clear your junk files and make sure you do your cookies to.
It should then come back.
4. you could have come in when the server was offline it doesn't happen often but it does from time to time.
The only time it was really bad was well when most services just stopped for an hour with no explanation.
It turned out from my isp that a local datacentre had lost one of their storage arrays and all drives in that and had to transfer to the us while they replaced it and then back again.