Define the function $f: \R \rightarrow \R$ by

$f(x) = \frac{1}{x^2 + \sqrt{x^4 + 2x}}$ where $x \not \in (-\sqrt[3]{2}, 0]$, and $f(x) = 0$ otherwise.

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, $d > 0$, $c$ is square-free, and $gcd(a,b,d) = 1$.

Find $1000a + 100b + 10c + d$

(Here, $f^n(x)$ is the function $f(x)$ fed into itself $n$ times, i.e. $f^3(x) = f(f(f(x)).)$

Source: HMMT November 2021, General Round

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