ÖMO 2003 Problem 2

Let $a, b, c$ be nonzero real numbers for which there exist $\alpha, \beta, \gamma \in\{-1, 1\}$ with $\alpha a + \beta b + \gamma c = 0$. What is the smallest possible value of $$\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?$$