
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 in the plane $x+y+z=1.$ Define that $(x,y,z)$ supports $(a,b,c)$ if exactly two of $x\ge a$, $y\ge b$, $z\ge c$ are true. Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ supporting $\left(\frac 13,\frac 14,\frac 15\right)$. Find the ratio of the area of $\mathcal{S}$ to the area of $\mathcal{T}$.

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