4. As shown in the figure, the radius of semicircle $O$ is 1, $A C \perp A B$ at $A, B D \perp A B$ at $B$, and $A C=1, B D=3, P$ is any point on the semicircle, then the maximum value of the area of the closed figure $A B D P C$ is
4. $2+\sqrt{2}$