Let $a,b,c$ be 3 distinct positive real numbers such that abc = 1. Prove that $$\frac{a^3}{\left(a-b\right)\left(a-c\right)}\ +\frac{b^3}{\left(b-c\right)\left(b-a\right)}\ +\ \frac{c^3}{\left(c-a\right)\left(c-b\right)}\ \geq 3$$
I tried AM-GM in many different ways, but it doesn't work since one of the terms on the LHS inevitably becomes negative. Any help is greatly appreciated.
By AM-GM $$\sum\limits_{cyc}\frac{a^3}{(a-b)(a-c)}=-\sum\limits_{cyc}\frac{a^3}{(a-b)(c-a)}=-\sum\limits_{cyc}\frac{a^3(b-c)}{\prod\limits_{cyc}(a-b)}=a+b+c\geq3$$