
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 sequence $\{a_n\}$ is an arithmetic sequence with $a_1 = 3$ and $a_4 = 12$, and the sum of the first $n$ terms of sequence $\{b_n\}$ is $S_n$, where $2S_n = 3^{n+1} - 3$ for all positive integers $n$.

(1) Find the general term formulas for both sequences $\{a_n\}$ and $\{b_n\}$.
(2) Let $\{c_n\} = \{b_n\} + \frac{9}{a_n a_{n+1}}$. Find the sum $T_n$ of the first $n$ terms of sequence $\{c_n\}$.

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