Inputting the password from the live system should allow you to access your encrypted data. But you can perform as well the backup from the text session you get with Ctrl+Alt+F1 (or Ctrl+Alt+F2 or...). For example, you can plug an external hard drive and copy /home there ('mount' tells where its filesystems are mounted) with 'sudo cp -a /home'. You can also diagnose the problem from the text session (using 'journalctl') and maybe fix it.

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