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.

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