Let $p(x)$ be a polynomial of degree strictly less than $100$ such that it does not have $(x^3-x)$ as a factor. If \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \) for some polynomials $f(x)$ and $g(x)$, then find the smallest possible degree of $f(x)$.

Let's think step by step and output the final answer within \boxed{}.
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