
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 pairs of complex numbers \((x, y)\) satisfying the equations
$$
x + 2y^2 = x^4 \quad \text{and} \quad y + 2x^2 = y^4,
$$
compute the minimum possible real part of \(x\).

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