
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 \( f_{1}(x) = \frac{2}{x+1} \) and \( f_{n+1}(x) = f_{1}\left[f_{n}(x)\right] \) for \( n \in \mathbf{N}^{*} \). Let \( a_{n} = \frac{f_{n}(2)-1}{f_{n}(2)+2} \). Find \( a_{99} \).

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