
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Define the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) by
\[ 
f(x)=\left\{\begin{array}{ll}
\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text{ if } x \notin (-\sqrt[3]{2}, 0] \\
0 & \text{ otherwise }
\end{array}\right.
\]

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 \( 1000a + 100b + 10c + d \).

(Here, \( f^{n}(x) \) is the function \( f(x) \) iterated \( n \) times. For example, \( f^{3}(x) = f(f(f(x))) \).)

Present the answer in LaTex format: \boxed{Your answer}
user