
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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Given a sequence $\{a_{n}\}$ that satisfies $a_{1}=1$, $a_{n+1}=a_{n}+2$ for all $n\in\mathbb{N}^{*}$, and a geometric sequence $\{b_{n}\}$. It is known that $a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+…+a_{n}b_{n}=(n-1)\cdot 3^{n+1}+3$.

1. Find the general term formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
2. If $a_{n}\cdot(1+2\log_{3}b_{n})\cdot c_{n}=1$, find the sum of the first $n$ terms $T_{n}$ of the sequence $\{c_{n}\}$.

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