For each $x$ in $[0,1]$ define $f(x)$ as shown in the image. Let $f^{[2]}(x)=f(f(x))$, and $f^{[n+1]}(x)=f^{[n]}(f(x))$ for each integer $n\ge 2$. For how many values of $x$ in $[0,1]$ is $f^{[2023]}(x)=\frac{1}{4}$?