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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