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.

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