Somersaults are more fun than this sum assault

Calculus Level 1

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

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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