Find the number of pairs $(z_1,z_2)$ of complex numbers such that:

$z_1 z_2$ is pure imaginary
$\frac{z_1}{z_2}$ is real
$|z_1| = |z_2| = 1.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the number of pairs $(z_1,z_2)$ of complex numbers such that:

$z_1 z_2$ is pure imaginary
$\frac{z_1}{z_2}$ is real
$|z_1| = |z_2| = 1.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
Find the number of pairs $(z_1,z_2)$ of complex numbers such that:

$z_1 z_2$ is pure imaginary
$\frac{z_1}{z_2}$ is real
$|z_1| = |z_2| = 1.$
train