More Wrecked Angles

Geometry Level 3

In the diagram above, ABCD and PQRS are both rectangles. Points P, Q, R, and S lie on segments AB\overline{AB}, BC\overline{BC}, CD\overline{CD}, and DA\overline{DA}, respectively, and BQ<QC\overline{BQ} < \overline{QC}.

If AB=36AB=36 and BC=50BC=50, then the maximum possible value of BQBQ can be written in the form aba-\sqrt{b}, where a,bNa,b \in \mathbb{N}. What is a+ba+b?

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