In a triangle $$ABC, \angle BAC = 90^\circ$$. Let $$D$$ be the point on $$BC$$ such that $$AB + BD = AC + CD$$. Suppose $$BD : DC = 2 : 1$$. If $$\frac{AC}{AB} = \frac{m+\sqrt{p}}{n}$$, where $$m, n$$ are relatively prime positive integers and $$p$$ is a prime number, determine the value of $$m + n + p$$.