
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 \( F_{1} \) and \( F_{2} \) be the foci of the hyperbola \(\Gamma: x^{2}-\frac{y^{2}}{3}=1 \). \( O \) is the origin, and point \( P \) lies on \(\Gamma\) such that \( |O P|=2 \). Determine the area of \(\triangle P F_{1} F_{2}\).

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