
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 \( A \) and \( B \) be the endpoints of the major axis of an ellipse \( \Gamma \), and let \( E \) and \( F \) be the foci of \( \Gamma \). Given that \( |AB|=4 \) and \( |AF|=2+\sqrt{3} \), and let \( P \) be a point on \( \Gamma \) such that \( |PE| \cdot |PF|=2 \). Determine the area of \( \triangle PEF \).

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