It will all click together

Geometry Level 5

Suppose a cyclic quadrilateral ABCDABCD is such that

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

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

(iii) cos(BAD)=13.\cos(\angle BAD) = -\dfrac{1}{3}.

Let RR be the radius of the circle in which ABCDABCD is inscribed. Find 1000R\lfloor 1000*R \rfloor.

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