We feel that anyone who experienced the Probe Module puzzle in Tournament 2023 must believe that this puzzle can be solved at P4. But just like most of the people who experienced that puzzle, we were unable to find a solution like that.

In fairness, we could argue that this puzzle is actually the more difficult if the two.  The edge of a hexagon is more restrictive than the wall of a cabinet. This puzzle requires constructing a large output, which is generally more geometrically challenging than deconstructing a large input.  And perhaps most importantly, this hexagon just has less area than that cabinet. 

Still, we did find fragments of a solve that got tantalizingly close to P4 or P5. The most promising of them got stuck in the "rocking" motion with the pivot and then translation, and no way to extract the finished product.  In order to make headway, we finally had to give up and just solve the puzzle.  

So we posted a P11 that we quickly improved to P9.  The main thing that was limiting us was the sheer space required to get multiple arms working on slow-timed fractional tape loops. An overnight inspiration got us to this idea, which only has one arm doing timing tricks, and the rest doing simple, every-loop geometry conditionals.

We are we are still 100% certain that better P is possible.  But given how challenging this puzzle is, we declare P6 to be a respectable bestagon.  A Respectagon, if you will.