Purposely or purposefully – which one is it?

My kids make this mistake so I thought a clarification might be helpful for others.
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According to the Grammarly blog, when you use purposely in a sentence, it should be synonymous with intentionally and on purpose. The opposite of purposely is accidentally.

When you use purposefully in a sentence, it should mean “in a way that shows determination and resolve” or, put more simply, “full of purpose.” The opposite of purposefully would be close to negligently or carelessly.

Click on the link for examples and more.

purposely or purposefully?