Subtlest hint:
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×Work backwards.
If you're not sure what that means, think about what sort of event you could be working backwards from.
Less subtle:
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×Check the move order of the lemmings.
More detailed:
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×Push the mirror onto the plate on the same turn you step on the disarm token. On this turn, all the lemmings will move, and you want the mirror to hit the other plate by the end of the turn.
So, start with the lemming that's last in the turn order and figure out where it has to go. Then take the lemming before it, figure out where it has to go, and block it with a decoy sword. If you just take it one lemming at a time, it's not too hard to work out.
