A group of \(x\) painters split up into \(y\ (\le x)\) groups of not necessarily the same size to work on a project of painting circles on the wall. Each painter is given just enough paint to fill in 4 square meters of any shape.

Now, if each group uses their total amount of paint to fill in a largest circle possible, and the sum of the circumferences of all these circles is at most \(k \sqrt{xy}\) meters with \(k\) a positive real number, what is \( \big\lfloor k^2 \big\rfloor?\)

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