
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 common ratio of a geometric sequence $\{a_{n}\}$ is $q=2$, and $a_{2}+a_{3}=12$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ If $\{b_{n}\}$ is an arithmetic sequence, and $b_{3}=a_{3}$, $b_{7}=a_{4}$, find the sum $T_{n}$ of the first $n$ terms of $\{b_{n}\}$.

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