MEMO 2013 I1

Let $a, b, c$ be positive real numbers such that $$a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .$$ Prove that $$2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } .$$ Find all triples $(a,b,c)$ for which equality holds.