
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 the sequence \(\left\{a_{n}\right\}\) that satisfies:
\[
\begin{array}{l}
a_{1}=\frac{9}{4}, \\
2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0 \text{ for } n \geq 1.
\end{array}
\]
(1) Let \(c_{n}=a_{n}-2\). Find the general term formula for the sequence \(\left\{c_{n}\right\}\).  
(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\), and define \(b_{n}=\frac{n^2}{n+1} a_{n}\). Find the largest positive integer \(n\) such that \(\sum_{k=1}^{n}\left[b_{k}\right] \leq 2019\).

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