MK-MO 2013 Problem 4

Let $a,b,c$ be positive real numbers such that $a^4+b^4+c^4=3$ . Prove that $$\frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6$$