
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
Given that \( \mathbf{R}^{+} \) is the set of all positive real numbers, and \( \alpha, \beta \) are given non-zero real numbers, find all functions \( f: \mathbf{R}^{+} \rightarrow \mathbf{R} \) such that for all \( x \) and \( y \) in \( \mathbf{R}^{+} \), the following inequality holds:
\[
f(x) f(y) \geq \frac{y^{\alpha}}{x^{\beta} + x^{\beta}} [f(x)]^{2} + \frac{x^{\beta}}{y^{\alpha} + y^{\beta}} [f(y)]^{2}.
\]

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