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I would not completely rule out the PSU being too weak. However, I would first see if cooling the system down would help(generally I don't see random shutdowns from a weak PSU). I do this at home by pulling the computer out into an open area(that is safe for it to run in) and then opening the case up and have a couple fans blowing cool air past it(note: not into it, you just want to keep as much 'warm' away from the normal fans as possible).
If it still shuts down randomly than you may conclude that the PSU might be too weak(it may be uncommon but I have seen a weak PSU cause random shutdowns once in my life).
As for keeping it cooler. Keep these in mind:
-You want easy airflow around the case.
-Don't put it towards one of your furnace registers(hot air being spat out right beside the computer = bad).
-Make sure to keep the insides and fans clean of dust. You would be surpised how much dust can increase a computer's temperature.
-Make sure you have air circulation in the room it is in(dead air doesn't really help it, though it doesn't exactly hurt it either if the air is cool).
As for the stated watts required for hardware... I tend to observe that you don't always need the 'recommended' watts to actually use the card reliably. I hear that can be awfully hard on your PSU though.
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