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Calculus Level 5

A circle of radius \(r \lt 1\) is internally tangent to a unit circle. A chord is drawn inside the unit circle that is tangent to the radius \(r\) circle and perpendicular to the line joining the centers of the two circles.

Let \(O\) be the center of the smaller circle, and let \(P, Q\) be the endpoints of the chord as described above. The value of \(r\) that maximizes the perimeter of triangle \(OPQ\) can be written as \(\dfrac{a - \sqrt{b}}{c}\), where \(a, b, c\) are positive coprime integers and \(b\) is square-free. Find \(a + b + c\).

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