
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
For a geometric sequence $\{a_{n}\}$ where all terms are positive, the first term is $1$. The sum of the first $n$ terms is denoted as $S_{n}$, and $a_{2}+S_{3}=16$. If the sequence $\{b_{n}\}$ satisfies $a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}=n\cdot 3^{n}$, then $b_{n}=$____.

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