HMMT 2011 S4

Let $a, b$, and $c$ be complex numbers such that $|a|=|b|=|c|=1$. If

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

as well, determine the product of all possible values of $|a+b+c|$.