Did you de-authorize the ESRI software before re-imaging the laptop? If not, you lost that license authorization, but ESRI does let you re-authorize one more time. After 2 license losses (at least in my experience), you need ESRI tech support to help out with the authorization.
The host 'not_set' error would indicate to me that you have your licensing set to Concurrent Use in the ArcGIS Administrator and have not set the licensing server:
As Kory already indicated, this error message means that the ArcGIS for Desktop software is set to utilize Concurrent licensing instead of Single Use licensing. I would first confirm via licenses listed at MyEsri whether you want to utilize Concurrent Use or Single Use for your licensing method.
Once that is confirmed, through the ArcGIS Administrator, you can set the radio button (the six options are shown in the image above) to the corresponding licensing method and level. If it is determined that you are to use Concurrent Use for licensing and you're not familiar with the name of the License Manager, I would reach out to your IT personnel or GIS Admin to figure out the name as this parameter is currently set to "Not_Set". The name of the License Manager can be entered within the circled box in the screen shot above.
Please reach out if you have additional questions or concerns.