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Math·Geometry and Trigonometry·Circles

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A circle has center $G$, and points $M$ and $N$ lie on the circle. Line segments $MH$ and $NH$ are tangent to the circle at points $M$ and $N$, respectively. If the radius of the circle is $247$ millimeters and the perimeter of quadrilateral $GMHN$ is $5,174$ millimeters, what is the distance, in millimeters, between points $G$ and $H$?

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