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Geometry Level 4

Let \(k\) be the ratio of the area of an ellipse \(A_e\) to the area of the triangle \(A_t\) the ellipse is inscribed in; that is: \[k=\dfrac {A_e} {A_t} .\]

Given that the maximum value of \(k\) has a closed form, find this closed form.

Submit your answer to 2 decimal places.

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