A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter \(10\) and altitude \(12\), and the axes of the cylinder and cone coincide. Find the radius of the cylinder. The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).

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