HMMT November 2021 E7

Define the function $f: \mathbb{R} \rightarrow \mathbb{R}$ by

$$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$

The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$. (Here, $f^{n}(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^{3}(x)=f(f(f(x)))$.)