My algebra has failed me!

Algebra Level 4

Consider two complex numbers $w$ and $z$. Suppose that these two numbers satisfy

$\begin{cases} |w|=|z| \\ |w+z|=|w-z| \end{cases}$

Then this implies that $w=\pm k \times z$.

What is $|\text{Im}(k)|+|\text{Re}(k)|$?

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