
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 geometric sequence $\{a_n\}$ satisfies $a_1=1$ and $2a_3=a_2$  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ If the sum of the first $n$ terms of an arithmetic sequence $\{b_n\}$ is $S_n$, satisfying $b_1=2$ and $S_3=b_2+6$, find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n \cdot b_n\}$.

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