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If you push down on the agitator and it works, it either means that your transmission is getting into gear or the agitator/hub is striping out. You'll need to remove the agitator and inspect to see if anything is stripped out below the agitator. These agitators are actually very difficult to remove. They are only held on by friction, but they do just pull straight up. GE has a tool which is similar to a bicycle inner tube which slides under the agitator and inflates to push the agitator up and off. Since you don't have that tool, you would likely need 2 people to just grab under the bottom of the agitator and pull up really hard.
If the agitator doesn't looked stripped out once you remove it, you'd need a new transmission which is about $300 to have replaced and maybe not worth fixing but that is up to you.
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