
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 $S_n=2a_n-2$. In the sequence $\{b_n\}$, $b_1 =3$, $b_4 =9$, and it satisfies $b_n+2+ b_2=2b_n+1$ for $n\in\mathbb{N}^*$. 

$(1)$ Find the general formula for the sequence $\{a_n\}$.

$(2)$ Let $c= \frac{b_n}{a_n}$, 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}
user