HMMT 2006 U22

Let $f(x)$ be a degree 2006 polynomial with complex roots $c_{1}, c_{2}, \ldots, c_{2006}$, such that the set

$$\left\{\left|c_{1}\right|,\left|c_{2}\right|, \ldots,\left|c_{2006}\right|\right\}$$

consists of exactly 1006 distinct values. What is the minimum number of real roots of $f(x)$ ?