
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 vertices of the major axis of ellipse \( \Gamma \). \( E \) and \( F \) are the foci of \( \Gamma \). Given that \( |AB| = 4 \) and \( |AF| = 2 + \sqrt{3} \). Point \( P \) lies on \( \Gamma \) and satisfies \( |PE| \cdot |PF| = 2 \). Find the area of \( \triangle PEF \).

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