Reset the activation count for Adobe CS 5.5 Design Premium

An impossible request. 

Nobody can reset CS activations anymore. That legacy technology no longer exists. 

CS5 REPLACEMENTS:
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FREE Acrobat Reader
https://www.adobe.com/acrobat/pdf-reader.html

FREE Online PDF Editor
https://www.adobe.com/acrobat/online/pdf-editor.html\

Paid Acrobat (Standard, Pro or Studio)
https://www.adobe.com/acrobat/pricing/compare-versions.html

Photoshop Elements (3 year license).
https://www.adobe.com/products/photoshop-elements.html

Creative Cloud Annual Photography plan. 
https://www.adobe.com/creativecloud/photography.html

FREE or Paid Adobe Express
https://helpx.adobe.com/express/using/express-overview.html

Creative Cloud Bundles & Single Apps for mobile, web & desktops
https://www.adobe.com/creativecloud/plans.html

Hope that helps.

Much like the old computer, CS5.5 is past its shelf life. Adobe is no longer able to manually deactivate the 15 year old software. If you are unable to deactivate the existing licenses to enable new activations your only option is to subscribe to the latest releases or consider alternatives.

deactivating can only be done using computers with a current activation. there's no alternative.