Intersection of circles

Geometry Level 4

Circles $$\Gamma_1$$ and $$\Gamma_2$$ have centers $$X$$ and $$Y$$ respectively. They intersect at points $$A$$ and $$B$$, such that angle $$XAY$$ is obtuse. The line $$AX$$ intersects $$\Gamma_2$$ again at $$P$$, and the line $$AY$$ intersects $$\Gamma_1$$ again at $$Q$$. Lines $$PQ$$ and $$XY$$ intersect at $$G$$, such that $$Q$$ lies on line segment $$GP$$. If $$GQ = 255$$, $$GP = 266$$ and $$GX = 190$$, what is the length of $$XY$$?

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