
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
Let the sequence $\{a_n\}$ be an arithmetic sequence, and the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$, satisfies $S_n=2(b_n-1)$, and $a_2=b_1-1$, $a_5=b_3-1$.  
(I) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(II) Let $c_n=a_n\cdot b_n$, 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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