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I'd first try removing the auxiliary drive belt and turning every pulley and ancillary ( power steering pump , alternator etc) by hand. All should turn easily (with the exception of the AC pump) with no grittyness or wobbly pulleys- If you do find a sticky pump / alternator then it should be replaced or repaired
If the engine is stone cold you can briefly run it with no belts to see if the noise changes or has gone away, this will help show if an ancillary is the issue
another trick to shut up a water pump ( if this is the culprit) is to add a teaspoon of brake fluid into the coolant as this dissolves in water and will add some lubricity to the seals which can shut things up
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