
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 one of the following is true: $x \ge a$, $y \ge b$, $z \ge c$. Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\right)$. Find the area of $\mathcal{S}$ divided by the area of $\mathcal{T}$.

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