\[ \large \dfrac{x^2}{100} + \dfrac{y^2}{225} = 1 \]

A right cylinder whose axis is along the \(z\)-axis, has an elliptical cross-section (in the horizontal plane) given above.

If a plane whose equation is \( \sqrt{2} x + \sqrt{2} y + 2 \sqrt{3} z = 100 \sqrt{3} \), cuts through the cylinder, then the cutting plane and the cylinder intersect in an ellipse. Find the sum of the semi-minor and semi-major axes of this ellipse.

Round your answer to three decimal places.

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