
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, satisfying $S_n=2-a_n$ $(n\in\mathbb{N}^*)$. The sequence $\{b_n\}$ satisfies $(2n-1)b_{n+1}-(2n+1)b_n=0$ $(n\in\mathbb{N}^*)$, and $b_1=1$.  
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $c_n=a_n\cdot b_n$, find the sum of the first $n$ terms of sequence $\{c_n\}$, denoted as $T_n$.

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