
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
The inverse function of \( f(x) \) is \( y = \frac{x}{1+x} \). Given that \( g_{n}(x) + \frac{1}{f_{n}(x)} = 0 \), let \( f_{1}(x) = f(x) \), and for \( n > 1 \) and \( n \in \mathbf{N}^{*} \), \( f_{n}(x) = f_{n-1}[f_{n-1}(x)] \). Find the analytic expression for \( g_{n}(x) \) for \( n \in \mathbf{N}^{*} \).

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