<s>[INST] What's the maximum value of $\frac{2}{a^2+1}-\frac{2}{b^2+1}+\frac{3}{c^2+1}$ given that: $abc+a+c=b$ 
Given that: $abc+a+c=b$. What's the maximum value of $$\frac{2}{a^2+1}-\frac{2}{b^2+1}+\frac{3}{c^2+1}.$$

(DO NOT use trigonometric methods)
 [/INST] prove that $$\frac{2}{a^2+1}-\frac{2}{b^2+1}+\frac{3}{c^2+1}\le \frac{10}{3}$$ and the equal sign holds if $$a=\frac{2}{3}\left(\frac{1}{2\sqrt{2}}-\sqrt{2}\right),b=-\sqrt{2},c=-\frac{1}{2\sqrt{2}}$$
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