A cone with vertex at the origin and opening upward, with axis along the positive \(z\)-axis, is scaled (stretched) along the \(y\)-axis direction by a factor of 2. The angle between the surface of the unstretched cone and its axis, is \( 20^{\circ} \). A plane with equation \( \sqrt{2} x + \sqrt{2} y + 2 \sqrt{3} z = 100 \sqrt{3} \) cuts though this stretched cone. The intersection between the plane and the stretched cone is 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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