Save the last successful backup and add --last-backup

Write the completion time and backup directory to
$XDG_STATE_HOME/simple_backup/last_backup after each successful backup
(as the target user when run as root), for status bar reminders.
--last-backup prints it.

tests/test_local.py

@@ -201,7 +201,11 @@ class TestRsync:
         for _ in range(2):
             assert run_cli('-c', config_file, '-i', str(src), '-o', str(out), '-k', '0') == 0
 
-        assert backups(out) == ['2026-10-14_11-00-00']
+        # The fake clock also advances when the state file is written
+        assert backups(out) == ['2026-10-14_12-00-00']
+        state = (tmp_path / 'state' / 'simple_backup' / 'last_backup').read_text().splitlines()
+        assert state[0].startswith('2026-10-14T13:00:00')
+        assert state[1] == f'{out}/simple_backup/2026-10-14_12-00-00'
 
 
 class ChunkStream(io.BufferedIOBase):