
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
Over all real numbers \( x \) and \( y \) such that
\[ x^3 = 3x + y \quad \text{and} \quad y^3 = 3y + x \]
compute the sum of all possible values of \( x^2 + y^2 \).

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