CAPS 2007 Problem 6

Let $ABCD$ 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 $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$