Traditionally presents are kept secret in boxes wrapped in paper. Since the ice caps are melting Santa Claus has begun to investigate some ways they might make the whole gift wrapping operation a little more eco-friendly. The first thing they've decided to do is to put all presents in perfectly cubic boxes. This makes things more efficient to process and cuts down on waste. The second thing they are doing is to reuse some of the scrap paper to wrap boxes as opposed to tossing it.
In this challenge you will help Santa make a machine which will identify which pieces of scrap can still be used to wrap gifts. Because of the industrial wrapper cutting machine used scraps will always be connected faces of a square tiling. The squares will also always be the exact size of the faces that make up the boxes Santa uses.
For example you might get the following scrap:
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This scrap can fold around the cube perfectly with no overlap and no empty space. Some scraps might overlap, for example:
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Can wrap around the cube perfectly, but since there are more than 6 squares there must be overlap. This is fine and this scrap can be used to wrap.
Some scraps can't wrap a cube without leaving some gaps. For example
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This has 6 squares, but there's no way to fold them which covers all sides of a cube. So this should be sent to the recycling plant.
Some scraps can wrap the cube but require fancy folding techniques. For example:
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This can be folded to wrap around a cube, but it requires folds that aren't along the edges of the squares. The wrapping machine just isn't that fancy, so this should be sent to the recycling plant.
The following can be folded with the only permanent creases being along the edges of the squares:
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However it requires making multiple folds at once and bending the paper to make the geometry fit. The machine can't do anything that fancy, it makes one fold at a time along one straight axis, folding every edge that meets that axis.
Task
For this challenge, take a scrap as input in any reasonable method. Output one of two consistent distinct values. One value should only be output if the scrap can be used as gift wrapping, the other should only be output if the scrap should be sent to the recycling plant.
Some folds are almost possible, but require paper to pass through itself to complete the fold. If the a shape requires this to fold you can output either value. That is you can choose to assume that paper can pass through itself or not, whichever is more convenient for you.
In order to be eco-friendly Santa is asking you to minimize the size of your source code as measured in bytes. (You've tried to explain that that's not how things work.) So this is code-golf and the goal is to minimize the size of your source code as measured in bytes.
Test cases
Truthy
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Falsy
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