A Beautiful but Difficult Inequality

Algebra Level 5

\[\dfrac{x}{\sqrt{x+y}}+\dfrac{y}{\sqrt{y+z}}+\dfrac{z}{\sqrt{z+x}}\]

Over all non-negative reals \(x, y, z \) satisfying \( x + y + z = 4 \), let the maximium value of the above expression be \( M \). What is the value of \( \lfloor 10000 \times M \rfloor \)?

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