Before disassembly, remove the aerator in the spigot. If you can't do so by hand, it may have caked-on dirt and grime. Dip one of your rags in white vinegar and secure it around the spigot with a rubber band for about an hour. Use the rag as a buffer to the wrench to unscrew the cap and take out the aerator. If it's dirty, you can drop it in some white vinegar, or you can just replace it.
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