Why will my 2002 Honda Accord not shift into park?

I can literally shift it into the park position, but it's not acting like it is in park. Even when I have it "shifted" to park, I can't start the car. I can only start it in Neutral.
When it is in the park position, the park light doesn't come on, and I can easily shift it down to reverse or neutral without having to press and hold the button. What could be wrong?

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