Seeking advice for fixing my gaming PC

I have an AlienWare desktop running Windows 8 and it has stopped working after 3 years, after its warranty has expired. The problem first occurred when my computer suddenly froze during normal use about a week ago. After a forced shut down, I'd turn it on again. The light would go on and I'd hear the machine starting up, and then it would turn off after about 3 seconds, and then it would start back up again after about 5 seconds. Then it would freeze, sometimes during boot, sometimes after Windows already had time to open a browser and Word.

When I tried it the next day it seemed to fix itself. I ran a Windows Defender scan and it didn't detect any malware. But now the problem has come back.

I thought it might have something to do with dust buildup, perhaps overheating. It looks like dust can get in through the top, so I tried blowing really hard through the holes on each side. I wanted to open it up to clean it, but I only see one screw on the back and removing it didn't seem to let me take the panel off.

I called AlienWare support and he suggested a Pre-boot System Assessment, but the computer seemed to freeze even during the assessment, which suggests hardware failure rather than a Windows problem. He doesn't think it's overheating because then the computer would just turn off rather than freezing. He said the motherboard might need to be replaced, and he suggested taking it to the Geek Squad for repairs.

I sort of have my doubts that the Geek Squad will be able to fix it and that I'll just be wasting my money. What do you think? Should I try to repair it or just buy a new computer?