Prove that for any $$$a, b, c \gt 0$$$ the following inequality is true: \begin{align*} \left(\frac{a^2+b^2+c^2}{3}\right)\left(\frac{b^3}{a}+\frac{c^3}{b}+\frac{a^3}{c}\right) \ \ge a(2b-a)+b(2c-b)+c(2a-c) \end{align*}