Kenmore icemaker problems

I had asked and got some great answers to some questions about a faulty ice maker. Our refrigerator is a Kenmore made by whirlpool that has been having problems with the ice maker. It fills and makes ice but every 3 or 4 sets of cubes it over fills and has to be removed and melted and reinstalled and then ok for 3 or 4 more times. So My question is do You think the problem is the ice maker or the valve that lets the water in and is there an easy way to check which it might be. Thanks for the help.

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