Consider the section \(S\) of the parabola \(y = 2 - x^{2}\) lying in the first quadrant. What is the minimum possible length \(L\) of a line segment (in the first quadrant) that is tangent to \(S\) and has one endpoint lying on the \(x\)-axis and the other on the \(y\)-axis?

If \(L = \sqrt{\dfrac{m + n\sqrt{n}}{128}}\) where \(m,n\) are positive integers with \(n\) square-free, then find \(m + n.\)

Comments: Happy Birthday, Parth Lohomi. :)

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