Geometry in inequality

Geometry Level 5

$\large{ \dfrac{2}{\sqrt{a^2+1}} + \dfrac{1}{\sqrt{b^2+1}} + \dfrac{1}{\sqrt{c^2+1}} }$

Suppose $a,b,c$ are positive real numbers satisfying $a+b+c = abc$ . Let $P$ be the maximum value of the above expression. Find the value of $\left\lfloor 10000P \right\rfloor$ .

Clarification: The numerator of the first fraction is indeed 2.