4. Let $A$ and $B$ be the vertices of the major axis of the ellipse $\Gamma$, $E$ and $F$ be the two foci of $\Gamma$, $|A B|=4,|A F|=2+\sqrt{3}$, and $P$ be a point on $\Gamma$ such that $|P E||P F|=2$. Then the area of $\triangle P E F$ is $\qquad$

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