
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 that the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n$, and $S_n=2a_n-2$, and $\{b_n\}$ is an arithmetic sequence, with $b_3=a_2$, $b_2+b_6=10$.  
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n(2b_n-3)\}$.

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