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Geometry Level 5

Suppose a cyclic quadrilateral \(ABCD\) is such that

(i) \(AB = AD = 1\),

(ii) \(CD = \cos(\angle ABC)\) and

(iii) \(\cos(\angle BAD) = -\dfrac{1}{3}.\)

Let \(R\) be the radius of the circle in which \(ABCD\) is inscribed. Find \(\lfloor 1000*R \rfloor\).

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