Well, your reasoning is correct, you just have to find a good strategy to understand these loop rooms.
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× A general hint that's good for all 3 solutions is that you must pretend to be the lemming and try to do some loops yourself. Don't worry about walls destroyed by rotating your sword, those are details that you can fix later.
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× Once you figured out the pattern of the pp that the lemming must follow it's pretty much done, you just have to be careful while breaking the walls.
For all 3 solutions there are a lot of routes, so it's hard to give hints.
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× For exit South solution the big problem is the northern plate, I suggest you to break few walls and use the mirror to help you. Also, outside walls might also be useful.
About the challenge:
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× If you'll find the same loop solution I did you'll see that it's very similar to exit South solution. There are several ways of doing it but in my solution I placed the mirror at (11,22).
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There's not some other world out there where everything's gonna be okay.
There's just this one, just this rock.
-33th Skywatcher
It's not efficient but since this is mostly about finding a path it should still be helpful.