
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 \(\alpha \neq 0\) is a real number, determine all functions \(f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}\) that satisfy

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

for all \(x, y \in \mathbb{R}_{>0}\).

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