
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 on the major axis of the ellipse $\Gamma$, $E$ and $F$ be the foci of $\Gamma$, $|AB|=4$, and $|AF|=2+\sqrt{3}$. Point $P$ is on $\Gamma$ such that $|PE||PF|=2$. Find the area of the triangle $\triangle PEF$.

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