I have no children, but I have two kitties. That means that I have two dreaded litter boxes. Have you ever caught on to the diaper genie concept? I have a small trash can beside the litter box. I line it with a shopping bag (making sure there are no holes in it). Then I add another. And another. And another. Until I have lined it with about a dozen such bags. Each day I pick out the litter box and then tie that bag. The next day I will pick out the litter box on top of the first bag. I empty it about every third day. It works well, keeps the box clean, and keeps me from having to run out the trash every single day.
For the sake of practicality, I do a similar thing with the bathroom trash can.... line it with several bags and then just tie and empty one about every third day. Keeps me from having to re-line the can every time.
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