
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 ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\). \( P \) is a point on the ellipse such that \(\left|P F_{1}\right| : \left|P F_{2}\right| = 2 : 1\). Find the area of triangle \(\triangle P F_{1} F_{2}\).

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