Somersaults are more fun than this sum assault

Calculus Level 2

n=01(4n)!\sum_{n=0}^{\infty} \frac{1}{(4n)!}

The above sum can be expressed in the form

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

where ii is the imaginary unit and nn is some positive multiple of 4.

Find mm.

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