BALKAN-SL 2018 A4

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that the following inequality holds:

$$2\left(a^{2}+b^{2}+c^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right) \geqslant 3(a+b+c+a b+b c+c a)$$

(Romania)