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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