Problem 473

8. The decagon $A B C D P Q R S T U V W$ has two parallel bases: square $A B C D$ and octagon $P Q R S T U V W$, all angles of which are equal, and eight lateral faces: triangles $A P Q, B R S, C T U, D V W$ and rectangles $D A P W, A B R Q, B C T S$ and $C D V U$. It is known that the area of the section of the decagon by a plane passing through points $D, Q$ and $S$ is $\frac{2299}{480},|A B|=\sqrt{2},|P Q|=1$. Find the distance between the bases of the decagon. Write the found value in the answer, rounding it if necessary.