12.1.14 Olympic geometry

Condition: The quadrilateral \( A B K D \) is inscribed in a circle \( \Omega \) of radius \( \sqrt{37} \). On the side \( K D \) a point \( C \) is chosen so that \( \angle B C D=90^{\circ} \). A circle \( \omega \) of radius 6, circumscribed around a triangle \( B C K \), is tangent to the segment \( A D \) and tangent to the line \( A B \). Find the length of the segment \( A B \), the angle \( B A D \) and the area of ​​the quadrilateral \( A B C D \).