
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $m+n.$ 

Present the answer in LaTex format: \boxed{Your answer}
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