
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
Given that point \( P \) lies on the hyperbola \( \Gamma: \frac{x^{2}}{463^{2}} - \frac{y^{2}}{389^{2}} = 1 \). A line \( l \) passes through point \( P \) and intersects the asymptotes of hyperbola \( \Gamma \) at points \( A \) and \( B \), respectively. If \( P \) is the midpoint of segment \( A B \) and \( O \) is the origin, find the area \( S_{\triangle O A B} = \quad \).

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