
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
Determine all functions \( f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} \) such that

\[ f(f(x) + y) = \alpha x + \frac{1}{f\left(\frac{1}{y}\right)} \]

for all \( x, y \in \mathbb{R}_{>0} \), where \(\alpha \neq 0\) is a real number.

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