ISL 2006 G8

Let $A B C D$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $$\angle P A B+\angle P D C \leq 90^{\circ} \text { and } \angle P B A+\angle P C D \leq 90^{\circ} \text {. }$$ Prove that $A B+C D \geq B C+A D$. (Poland)