
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 the sequence $\{a_n\}$ be defined as $a_1=7$, $a_2=3$, and $a_{n+1}=3a_n-2$ for $n\geq 2$.

(1) Find the general formula for the sequence $\{a_n\}$.

(2) If the sequence $\{b_n\} = \frac{a_n-1}{2}$ and the sequence $\{c_n\}$ satisfies $c_n=\log_3b_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\{c_nb_n\}$.

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