BALTIC-WAY 2001 Problem 8

Let $A B C D$ be a convex quadrilateral, and let $N$ be the midpoint of $B C$. Suppose further that $\angle A N D=135^{\circ}$. Prove that

$$|A B|+|C D|+\frac{1}{\sqrt{2}} \cdot|B C| \geqslant|A D|$$