
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 $\left\{a_{n}\right\}$ satisfying $a_{1}=\frac{9}{4}, 2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0\left(n \in \mathbf{N}_{+}\right)$.
(1) Let $c_{n}=a_{n}-2$, find the general term formula for the sequence $\left\{c_{n}\right\}$;
(2) Let $b_{n}=\frac{n^{2}}{n+1} a_{n}$, find the maximum positive integer $n$ such that $\left[b_{1}\right]+\left[b_{2}\right]+\cdots+\left[b_{n}\right] \leqslant 2019$.
(Here, the symbol $[x]$ indicates the greatest integer less than or equal to $x$.)

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