Seek the $k$

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

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.