
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
In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\).

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