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Your power supply provides power to your CPU cooling fan either directly from the power supply or through your motherboard connection.
If your fan is not working, a sinple test to see if your fan is getting power is to use a test meter. Connect one lead to each of the wires (use a paperclip to inset it in the connector) set the volt meter to DC volts (symbol is a straight dashed line for DC voltage. AC is the squigly line).
You should read 12 or 24 volts. It the voltage is there, your fan is bad. If the voltage is not there, check your test lead connections again. It if still not there then your source power to your fan is not present. This could be a power supply problem (or motherboard if connected through the motherboard).
Get some compressed air and blow out your power supply. This may solve your problem. If the power supply is still not giving you the power to your connector, check that the wires are no broken or frayed.
If none of these do the trick, you may want to consider a new power supply, or route new wires to your connector.
An easier and safer test is to swap the fan or plug the fan into another supply plug. Even giving it a flick with your finger may get it going, and thus prove power is present. Messing with big old test leads on a hot motherboard is really something a technician should do and a task he would do only with extreme care - one tiny slip or misplaced lead tip could do irreparable damage to your motherboard, CPU, and/or RAM.
Keeping the system clean is good advice - cleaning it out after it has overheated may be too late.
Nevertheless, heat is the bane of electronics - fans are cheap - bearings freeze - I recommend you don't use that computer unless and until you are certain the fan is cooling.