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)