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Fan not working, laptop overheating, then not responding. I did Apple Hardware Test (extended), and no troubles found. Manual says “the problem might be software related.” What can I do now?

Most of the time when a Mac overheats it does an auto-shutdown for self protection. I would open it, see if the fan is running, remove the fan and clean the vent, then test again. To help us with diagnostics please give us the last four figures of your serial number. To test for a software conflict or possible virus, hold down the Shift key on start up. Let us know if any of your results change. Also how much available free hard drive space do you have and what OS are you running?

I would download a program called smc fan control and see if you can manually set the fan speed to see if it is working but not getting told to turn on or if it just plan does not work! If not ifixit sells the replacement fan and has the guide! MacBook Pro 13" Unibody and MacBook 13" Unibody Fan

The more software you install, the more works the hardware should do, then, your laptop will become overheating. You need to uninstall some software that not commonly used. And you need to buy a radiator to help.