Let \mathcal{T} be the set of ordered triples (x, y, z) of nonnegative real numbers that lie in the plane x+y+z=1. Let us say that (x, y, z) supports (a, b, c) when exactly two of the following are true: x \geq a, y \geq b, z \geq c. Let \mathcal{S} consist of those triples in \mathcal{T} that support \left(\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\right). The area of \mathcal{S} divided by the area of \mathcal{T} is m / n, where m and n are relatively prime positive integers. Find m+n.