JBMO-SL 2011 A7

Let $a, b, c$ be positive real numbers with $a b c=1$. Prove the inequality:

$$\frac{2 a^{2}+\frac{1}{a}}{b+\frac{1}{a}+1}+\frac{2 b^{2}+\frac{1}{b}}{c+\frac{1}{b}+1}+\frac{2 c^{2}+\frac{1}{c}}{a+\frac{1}{c}+1} \geq 3$$