
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
Determine all functions \( f: \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \) for which the following applies:

\[ \frac{1}{x} f(-x) + f\left(\frac{1}{x}\right) = x \quad \forall x \in \mathbb{R} \setminus \{0\} \]

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