Triangle \(ABC\) is a right triangle with \(\angle ACB\) as its right angle, \(m\angle ABC = 60^\circ\), and \(AB = 10\). Let \(P\) be randomly chosen inside \(\triangle ABC\), and extend \(\overline{BP}\) to meet \(\overline{AC}\) at \(D\). Find the probability that \(BD > 5\sqrt{2}\). The original answer is in the form \(\frac{k - \sqrt{m}}{n}\), please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
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