Let $ABCD$ be a convex quadrilateral with $\angle DAB = \angle BDC = 90^\circ$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$, then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$?

Let's think step by step and output the final answer within \boxed{}.
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