HMMT 2013 U25

The sequence $\left(z_{n}\right)$ of complex numbers satisfies the following properties:

• $z_{1}$ and $z_{2}$ are not real.
• $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \geq 1$.
• $\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \geq 1$.
• $\left|\frac{z_{3}}{z_{4}}\right|=\left|\frac{z_{4}}{z_{5}}\right|=2$.

Find the product of all possible values of $z_{1}$.