Amusing concept. I quite like how random objects get generated from the hat! haha. It adds a lot. I found myself really wanting the ability to put objects back into the hat; I imagine this is something you tried though.
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I never actually really tried messing around with being able to put objects back into the hat, not because I never thought of it but because a fundamental part of the design for me was always the permanence of object creation and having to deal with the mess you made. Having that ability would trivialize a few situations in the game, but that's not to say a different experience couldn't be made with it.