JBMO-SL 2019 A2

Let $a, b, c$ be positive real numbers such that $a b c=\frac{2}{3}$. Prove that

$$\frac{a b}{a+b}+\frac{b c}{b+c}+\frac{c a}{c+a} \geqslant \frac{a+b+c}{a^{3}+b^{3}+c^{3}}$$