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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