AMC 12 2014a Problem 24

Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?

$\textbf{(A) }299 \textbf{(B) }300 \textbf{(C) }301 \textbf{(D) }302 \textbf{(E) }303$