HMMT 2011 U34

Let $w=w_{1}, w_{2}, \ldots, w_{6}$ be a permutation of the integers $\{1,2, \ldots, 6\}$. If there do not exist indices $i<j<k$ such that $w_{i}<w_{j}<w_{k}$ or indices $i<j<k<l$ such that $w_{i}>w_{j}>w_{k}>w_{l}$, then $w$ is said to be exquisite. Find the number of exquisite permutations.