In the circle with center \(O\) and radius \(r\), triangle \(ABC\) is inscribed such that \(AB\) and \(BC\) are chords of the circle. Let \(D\) be the point where the perpendicular from \(O\) to \(BC\) intersects \(BC\). Given that \(AB = 6\), \(BC = 8\), and \(BD = 4\), express the radius \(r\) in terms of variables and then solve the system of equations to find the value of \(r\). Let's think step by step and output the final answer within \boxed{}.
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