ICTM State 2015 Question 19

In the diagraam shown, $B$ lies on $\overline{AC}$ , $AB=12, BC=12, \angle ADB = 15^\circ$ and $\angle BDC = 30^\circ$ .

If the length of $AD$ can be expressed as $k + w \sqrt f$ , where $k,w$ and $f$ are positive integers with $f$ square-free, find $k+w+f$ .