
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 sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, $\log _{3}b_{n+1}-1=\log _{3}b_{n}$, and $2a_{n}=a_{n+1}+a_{n-1}$ for $n\geqslant 2$. It is also known that $S_{3}=b_{3}=9$ and $b_{4}=a_{14}$. Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$. If $c_{n}=a_{n}b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.

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