
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
Let \( f(x) \) be a polynomial function defined over the set of real numbers, such that \( f(0) = 1 \) and for any \( x \cdot y \in \mathbb{R} \), it satisfies
$$
f(x-y) + f(x) \geqslant 2x^{2} - 2xy + y^{2} + 2x - y + 2
$$

Determine the function \( f(x) \).

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