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The compressor fan (known as the condenser fan) should turn on the moment that the compressor turns on. Whenever the compressor is on the condenser fan motor should be running. if it's not running that definitely will cause the compressor to over work. Put a fan back by the compressor to cool the compressor and see if the unit cools down like normal. If not then the condenser fan motor next to the compressor failed and caused the compressor to overwork for too long causing the compressor to become inefficient so now you'd need a new fan and a new compressor to fix the issue. That's a $700 repair or so, so it would likely not be worth repairing.
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alright good luck to you