Seek the \(k\)

Algebra Level 4

\[\dfrac{xy}{\sqrt{(x^2+y^2)(3x^2+y^2)}} \leq \dfrac{1}{k}\]

For all the positive real numbers \(x, y\), find the greatest real number \(k\) such that the inequality above is fulfilled.

If \(k = a + \sqrt b\) for \(a\) an integer and \(b\) a positive integer, enter the value of \(a+b\) as your answer, or else insert 0.

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