HMMT 2016 U34

(Caos) A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns. An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{4}\right\rfloor$ points.