The coordinates of \(A\), \(B\), and \(C\) are \((5,5)\), \((2,1)\), and \((0,k)\) respectively. Find the value of \(k\) that makes \(\overline{AC} + \overline{BC}\) as small as possible. The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

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