For $n \geq 4$, evaluate

$\large i^{0!}+i^{1!}+i^{2!}+i^{3!}+\cdots+i^{n!}.$

**Clarification**:

- $i = \sqrt{-1}$.
- $n! = n\times(n-1)\times(n-2)\times\dots\times3\times2\times1$.

###### Inspired by the series *Called Complex For A Reason* by Zandra Vinegar. Try her Part I, Part II and Part III