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: $$x\ge a, y\ge b, z\ge c.$$ Let $$\mathcal{S}$$ consist of those triples in $$\mathcal{T}$$ that support $$\left(\frac 12,\frac 13,\frac 16\right).$$ 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 $$m+n.$$

(第十七届AIME1999 第8题)