
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 positive arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, where $S_{3}=12$. If $a_{2}$ is the geometric mean of $2a_{1}$ and $a_{3}+1$. <br/>$(1)$ Find the general formula for $\{a_{n}\}$; <br/>$(2)$ Let ${b_n}={a_n}•{3^n}$, and the sum of the first $n$ terms is denoted as $T_{n}$, find $T_{n}$.

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