
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
Consider all polynomials \( P \) with real coefficients that satisfy the following property:

\[ 
\left|y^{2} - P(x)\right| \leq 2|x| \text{ if and only if } \quad\left|x^{2} - P(y)\right| \leq 2|y| 
\]

Determine all possible values of \( P(0) \).

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