My algebra has failed me!

Algebra Level 4

Consider two complex numbers ww and zz. Suppose that these two numbers satisfy

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

Then this implies that w=±k×zw=\pm k \times z.

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

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