Suppose $z$ is a complex number such that $z^5 = -\sqrt{3} + i$ and $z = \sqrt[5]{2} ( \cos n^{\circ} + i \sin n^{\circ})$, where $270 < n < 360$. Find the exact value of $n$.

**Details and assumptions**

$i$ is the imaginary unit, where $i^2=-1$.

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