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$\sum_{n=0}^{\infty} \frac{1}{(4n)!}$

The above sum can be expressed in the form

$\frac{1}{n} \left( \sum_{k=1}^{mn} e^{i^{k}} \right)$

where $i$ is the imaginary unit and $n$ is some positive multiple of 4.

Find $m$.

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