
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 $x + y + z = 2.$ Define $(x,y,z)$ to support $(a,b,c)$ when at least two of the following conditions are met: $x \ge a$, $y \ge b$, $z \ge c$. Let $\mathcal{S}$ be the subset of $\mathcal{T}$ that supports $(1, \frac{2}{3}, \frac{1}{3})$. Find the area of $\mathcal{S}$ divided by the area of $\mathcal{T}$.

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