
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 \( S \) be the set of points \(\left(x_{1}, x_{2}, \ldots, x_{2012}\right)\) in 2012-dimensional space such that \(\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{2012}\right| \leq 1\). Let \( T \) be the set of points in 2012-dimensional space such that \(\max _{i=1}^{2012}\left|x_{i}\right|=2\). Let \( p \) be a randomly chosen point on \( T \). What is the probability that the closest point in \( S \) to \( p \) is a vertex of \( S \)?

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