<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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  $m+n.$  Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
25
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  $m+n.$ 
test