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Test your phone with this
https://play.google.com/store/apps/details?id=com.sonymobile.androidapp.functiontester
and repair your phone with SUS or PCC
Update Service (SUS)
http://www.sonymobile.com/gb/tools/update-service/
http://www-support-downloads.sonymobile.com/Software%20Downloads/Update_Service_Setup-2.11.12.5.exe
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Don't forget the Kudos
You have to take your phone to a Sony repair center
Take it to your carrier/network they will guide you if they can repair it or if you have to contact Sony directly