In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). 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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