
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 continuous real functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) that satisfy, for every positive real \( x \), the condition

$$
x + \frac{1}{x} = f(x) + \frac{1}{f(x)}
$$

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