
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
(The full score of this question is 13 points) Let the sequence $\{a_n\}$ satisfy: $a_1=1$, $a_{n+1}=3a_n$, $n\in\mathbb{N}^*$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$, given that $b_1\neq 0$, $2b_n-b_1=S_1\cdot S_n$, $n\in\mathbb{N}^*$.  
(I) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
(II) Let $c_n=b_n\cdot\log_3a_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, $T_n$.

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