First take reflection, then cyclic

Geometry Level 5

In figure below,

\(ABCD\) is a convex quadrilateral with \(\angle DAB = \angle BDC = { 90 }^{ o } \). Let the incircles of triangles \(ABD\) and \(BCD\) touch \(BD\) at \(P\) and \(Q\). If \(AD = 999\) and \(PQ = 200\), find the sum of the inradii of two in-circles.

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