Don't let you into chaos when you see a weird inequality

Algebra Level 4

\[a\sqrt[3]{8+b-c}+b\sqrt[3]{8+c-a}+c\sqrt[3]{8+a-b}\]

Let \(a,b\) and \(c\) be non-negative numbers satisfying \(a^2+b^2+c^2=\dfrac{1}{3}\).

Find the maximum value of the expression above.


Source: My school selection MO for grade 10.
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