\[ \large f(x) = x^4 e^{x} \]

If the \(100^\text{th}\) derivative of \(f(x) \) at \(x=1\) is equal to \(k e\) for some positive integer \(k\), find \(k\).

**Notation:** \(e \approx 2.71828\) denotes the Euler's number.

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