
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 \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \), and \( P \) be a point on the ellipse such that \( \left|P F_1\right|:\left|P F_2\right| = 2:1 \). Find the area of the triangle \( \triangle P F_1 F_2 \).

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