
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 $f$ which is not defined for $x = 0,$ but for all non-zero real numbers $x,$
\[f(x) + 2f \left( \frac{1}{x} \right) = x^3 + 6.\]
Find the real solutions to $f(x) = f(-x).$

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