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Your first approach should be asking your host to fix the PHP session save path to something which is writable by your hosting account. This is a very common misconfiguration issue on Plesk-based hosts. If your host can't fix it, they might be able to give you instructions for creating a .htaccess to override the PHP session save path. No matter which solution they provide, you don't have to run Kickstart again or upload the extracted files again; you can access ANGIE again by visiting http://www.example.com/installation/index.php where www.example.com is the domain name you are restoring to.
Another solution is to change the permissions of the installation/tmp directory after Kickstart has extracted your archive (just right before you click the button) or after you have uploaded the extracted files if you're not using Kickstart. In both cases, try using your FTP client to change the permissions of the installation/tmp directory (but NOT its contents!) to 0777. Then, click on the Run the Installer button in Kickstart or access ANGIE again.
If both of these solutions don't work out for you, you will unfortunately have to go through the scenic route, i.e. perform a manual restoration. The complete instructions are described in the "Unorthodox: the emergency restoration procedure" section of our User's Guide. Our suggestion: it's best to switch to a different host. There is a strong possibility that Joomla! might not work properly unless you have a working PHP session storage.