
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 a sequence $\{a_n\}$ that satisfies the conditions $a_1=1$ and $2a_{n+1}=2a_{n}+1$ for $n\in \mathbb{N}^*$, the sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, and $S_n=2^{n+1}-2$.  
(I) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.  
(II) Let $c_n=a_n\cdot b_n$ for $n\in \mathbb{N}^*$, 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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