Let $S = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^4}$ and let $S_N = \sum_{n=1}^N \frac{(-1)^{n-1}}{n^4}$ be the $N$-th partial sum. By the Alternating Series Estimation Theorem, what is the smallest integer $N$ such that the error $|S - S_N|$ is guaranteed to be less than $10^{-6}$?