# 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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