In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \(\cot \angle DBC\) form an arithmetic progression. The area of \(\triangle ABC\) is in the form \(\frac{m}{n}\). Please calculate the value of \(m+n\).

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