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Let $z$ be a complex number satisfying:

$\color{#69047E}{\huge{|z|=|z+1|=1}}$

Also,

$\color{#20A900}{\huge{z^{n}=(z+1)^{n}}}$ where $\color{#3D99F6}{n}$ is a positive integer.

Find the minimum value of $\color{#D61F06}{n}$.

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