Store everything in Super Extra Large bins - Rubbermaid makes some that are the bees' knees. Then:
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I've seen this trick used to keep tubs and bins shut on plane flights:
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Use a hole punch/awl to make two holes - one in each handle or side of the bins. Zip-tie one hole, so it functions like a hinge. For the other hole, use a locking clip or some type of tie or latch your kiddo doesn't have the dexterity to open. Allow out only the number of toys your child is capable of putting away solo. Once those toys are out, clip the bins shut. If your sweetie wants another toy out, she has to come get you - which means you can enforce a "one in, one out" policy, praise her for being helpful, and help her choose things she's capable of putting away. (I work with kids: they LOVE taking out big projects, but struggle to put all the itty bitty component parts away when they're done. Are all of her toys things that a child her age can REALISTICALLY put away on her own?)
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As she shows more ability to self-regulate with toys and projects, allow more "unclipped" access to the bins. If she backslides, decrease unclipped time and make sure she understands why.
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It works where I work, ymmv.