
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
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x \in \mathbb{R}^{*} \) (nonzero real numbers) and all \( y \in \mathbb{R} \),

$$
f\left(x^{2} + y\right) \geq \left(\frac{1}{x} + 1\right) f(y)
$$

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