
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 that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=-\frac{9}{4}$, and $4S_{n+1}=3S_{n}-9$. Find:<br/>
$(1)$ The general term formula of the sequence $\{a_{n}\}$;<br/>
$(2)$ Let $\{b_{n}\}$ be a sequence satisfying $3b_{n}+na_{n}=0$ for $n\in \mathbb{N}^{*}$. Denote the sum of the first $n$ terms of the sequence $\{b_{n}\}$ as $T_{n}$. If $T_{n}\leqslant \lambda b_{n}+12$ holds for any $n\in \mathbb{N}^{*}$, find the range of real number $\lambda$.

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