This is a strategy challenge. It's difficult to find the right path. And I couldn't really tell you which path is the solution even if I wanted to. ...
(13 years ago) I'd never think this was the necessary order, before trying it.
Nearly 90 tries.
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EDIT 2022, 28th Oct., I had this tab opened for a looong time. Seeing Diab's comment here, today, encouraged me to try it again. My own comment from the past helped me to make what I call ''a desperate try'' :) (trying a path, even when you feel like it's longer, for sure). And this was the magic wand.
GREAT level, indeed. Agree with Diablillo, again! :)