
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
Let $f_{1}(x) = \frac{2}{1+x}$ and $f_{n+1}(x)=f_{1}\left[f_{n}(x)\right]$, and let $a_{n} = \frac{f_{n}(0)-1}{f_{n}(0)+2}$ for $n \in \mathbf{N}^{*}$.
(1) Find the general formula for the sequence $\left\{a_{n}\right\}$.
(2) Given that
$$
\begin{aligned}
T_{2n} &= a_{1} + 2 a_{2} + 3 a_{3} + \cdots + 2n a_{2n}, \\
Q_{n} &= \frac{4n^{2} + n}{4n^{2} + 4n + 1},
\end{aligned}
$$
where $n \in \mathbf{N}^{*}$, compare the sizes of $9 T_{2n}$ and $Q_{n}$, providing an explanation.

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