I managed all of this with the snakes, but got stuck at the last bit, going round in circles. In case someone else has come here with the same problem, and, like me, doesn't really want to sit through a demo of the snakes to figure out what they're doing wrong with the last bit, the answer is...
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×Don't get the roach into the south corridor.
In case that's not hint enough...
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×The roach will stick on the south arrow, giving you that extra turn you need.