Geometry in inequality

Geometry Level 5

2a2+1+1b2+1+1c2+1\large{ \dfrac{2}{\sqrt{a^2+1}} + \dfrac{1}{\sqrt{b^2+1}} + \dfrac{1}{\sqrt{c^2+1}} }

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

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

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