
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: $a_1=1$, $2a_{n+1}=2a_n+1$, $n\in\mathbb{N}^+$, and the sum of the first $n$ terms of sequence $\{b_n\}$ is $S_n$, where $S_n= \frac{1}{2}(1- \frac{1}{3^n})$, $n\in\mathbb{N}^+$  
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let $c_n=a_nb_n$, $n\in\mathbb{N}^+$, find the sum of the first $n$ terms of sequence $\{c_n\}$, $T_n$.

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