Advanced Number Theory · Superchallenge problem

In triangle $ABC$, $\angle A=45^\circ$, $\angle B=60^\circ$, and the side opposite $\angle A$ has length $6\sqrt2$. The area of the triangle is $a+b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is squarefree. Find $a+b+c$.