Find the Maximum

Algebra Level 3

As \(a,b,c\) range over all positive reals less than \(1\) satisfying \[3abc = \dfrac{1}{\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}},\] find the maximum value of \[\dfrac{1}{a^2+1-bc} + \dfrac{1}{b^2+1-ca} + \dfrac{1}{c^2+1-ab}.\]

Details and assumptions
- This problem is not original.

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