I have a mac air and it says my star up disk is full, now when I log in it just has a blank white lit screen and I cannot go further

After logging in, theres just a blank white backlit screen and I cannot go further. It also has been showing on the screen
that the startup disk is full.  I already tried diagnostics and it says hardware is fine.

You will have to erase your drive because it has a corrupted directory resulting from allowing the drive to become full.
Install or Reinstall OS X from Scratch
Be sure you backup your files to an external drive or second internal drive because the following procedure will remove everything from the hard drive.
Boot to the Recovery HD:
Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.
Erase the hard drive:
  1. Select Disk Utility from the main menu and click on the Continue button.
  2. After DU loads select your startup volume (usually Macintosh HD) from the
      left side list. Click on the Erase tab in the DU main window.
  3. Set the format type to Mac OS Extended (Journaled.) Optionally, click on
      the Security button and set the Zero Data option to one-pass. Click on
      the Erase button and wait until the process has completed.
  4. Quit DU and return to the main menu.
Reinstall OS X: Select Reinstall OS X and click on the Install button.
Note: You will need an active Internet connection. I suggest using Ethernet if possible
           because it is three times faster than wireless.

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