6. Given the circle $x^{2}+y^{2}=4$ and the point $P(2,1)$, two perpendicular lines are drawn through point $P$ intersecting the circle at points $A, B$ and $C, D$ respectively, with point $A$ lying on the segment $P B$ and point $D$ lying on the segment $P C$. The maximum value of the area of quadrilateral $A B C D$ is $\qquad$ .
Given that the midpoints of $A B$ and $C D$ are $E$ and $F$ respectively,
$$
\begin{array}{l}
\text { then } S_{A B C D}=\frac{1}{2} P B \cdot P C-\frac{1}{2} P A \cdot P D \\
=\frac{1}{2}[(P E+E B)(P F+F C)-(P E-E B)(P F-F C)] \\
=P E \cdot F C+E B \cdot P F=O F \cdot F C+O E \cdot E B \\
=O F \cdot \sqrt{4-O F^{2}}+O E \cdot \sqrt{4-O E^{2}} \\
\leqslant \sqrt{\left(O F^{2}+O E^{2}\right)\left(4-O F^{2}+4-O E^{2}\right)} \\
=\sqrt{O P^{2}\left(8-O P^{2}\right)}=\sqrt{5 \times 3}=\sqrt{15},
\end{array}
$$

Equality holds when $O E P F$ is a square.