Spin the Segment

Geometry Level 4

Circles $$\Gamma_1$$ and $$\Gamma_2$$ have radii of 106 and 200, respectively, and they intersect at two distinct points, A and B. A line passing through A intersects $$\Gamma_1$$ at P and $$\Gamma_2$$ at Q. If $$AB=112$$, find the maximum possible length of $$PQ$$.

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