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Calculus Level 2

Let \(S\) be the maximum possible area of a right triangle that can be drawn in a semi-circle of radius \(1\), where one of the legs (and not the hypotenuse) of the triangle must lie on the diameter of the semicircle.

If \(S = \dfrac{a\sqrt{b}}{c},\) where \(a,c\) are positive coprime integers and \(b\) is a positive square-free integer, find \(a + b + c.\)

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