
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 \(f, g: (0, +\infty) \rightarrow (0, +\infty)\) such that for all \(x \in \mathbf{R}^{-}\), the following holds:
$$
f(g(x)) = \frac{x}{x f(x) - 2}, \quad \text{and} \quad g(f(x)) = \frac{x}{x g(x) - 2}.
$$

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