KöMaL 2023 A861

Let $f(x)=x^2-2$. Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of $f$ (i.e. $f^{(1)}(x)=f(x)$ and $f^{(k+1)}(x)=f\big(f^{(k)}(x)\big)$). Let $H=\big\{x\colon f^{(100)}(x)\le -1\big\}$. Find the length of $H$ (the sum of the lengths of the intervals in $H$). We expect the solution as a closed-form expression.