Geometry in inequality part II

Geometry Level 4

Suppose that \(x,y,z\) are reals positive number satisfying the conditional: \(xyz+x+z=y\). Let the maximum is \(P\) \[P=\frac{2}{x^2+1}-\frac{2}{y^2+1}-\frac{4z}{\sqrt{z^2+1}}+\frac{3z}{(z^2+1)\sqrt{z^2+1}}\] If \[P=\frac{m}{n}\] Find the value of \( m+n \).

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