
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]


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Let the sequence $\{a_n\}$ satisfy: $a_1=1$, $a_{n+1}=a_n+2$, where $n \in \mathbb{N}^*$, and let the sequence $\{b_n\}$ be a geometric sequence. It is known that $a_1b_1+a_2b_2+a_3b_3+\ldots+a_nb_n=(n-1) \cdot 3^{n+1}+3$.

(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

(2) Let $a_n \cdot (1+2\log_{3}b_n) \cdot c_n=1$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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