In a triangle $$ABC$$, a point $$P$$ in the interior of $$ABC$$ is such that $$\angle BPC-\angle BAC=\angle CPA-\angle CBA=\angle APB-\angle ACB$$. Suppose $$\angle BAC=30^{\circ}$$ and $$AP=\sqrt{m}$$. Let $$D, E, F$$ be the feet of perpendiculars from $$P$$ to $$BC, CA, AB$$, respectively. If $$\frac{m}{n}$$ is the area of the triangle $$DEF$$ where $$m, n$$ are integers with $$n$$ prime, then what is the value of the product $$mn$$?