AMC 12 2005b Problem 22

A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule

$$z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},$$

where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?

\textbf{(A)}\ 1 \textbf{(B)}\ 2 \textbf{(C)}\ 4 \textbf{(D)}\ 2005 \textbf{(E)}\ 2^{2005}