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1. An other solution. It is based on idea of Yashenko. This way you can incresae the perimeter just a bit, but it is done by repeating one fold (which is very simple but not "simple" in the sense below).

2. It is still not known if you can increase the perimeter by a sequence of simple natural folds; i.e., folds like this: I just learned that this problem also appears in Pak's book, Problem 40.16b; it is marked by [$*$] which means that the problem is open.

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