
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
If
$$
f(x) = \frac{1}{\sqrt[3]{x^{2}+2x+1} + \sqrt[3]{x^{2}-1} + \sqrt[3]{x^{2}-2x+1}}
$$
for all positive integers \( x \), find the value of
$$
f(1) + f(3) + f(5) + \cdots + f(997) - f(999).
$$

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