You don't need the snake to open both doors, you can have the roach open one and the snake open the other.
×But how do you get the roach to go down wither the top-most or bottom-most path? You can't just make a run for it at the start, as you won't hit time to hit the West-most orb without getting the roach stuck or going the wrong place, and if you leave the mirror at position 1, when you go down to hit the orb, the snake turns towards the roach and dies in the doorway.
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×What if it wasn't possible for the snake to turn down that first central door and die behind the roach?
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×Your mirror position is pretty close. Like really, really really close. Say, one grid square off.
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×Put the mirror one tile east, and have the roach sit in the doorway while you move to strike the West-most orb. Then you can safely manoeuvre the roach down the Southern-most path and kill him before the snake can lock you in.
