HMMT 2004 A7

If $x, y, k$ are positive reals such that

$3=k^{2}\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right),$

find the maximum possible value of $k$ .

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{-1 + \sqrt{7}}{2}

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