
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 sequence $\{a\_n\}$ is an arithmetic sequence, the sum of the first $n$ terms is $S\_n$, and it satisfies $a\_3 - a\_1 = 4$, $S\_3 = 12$.

1. Find the general term formula for the sequence $\{a\_n\}$;
2. Let $b\_n = a\_n \cdot 2^{n-1}$, find the sum of the first $n$ terms for the sequence $\{b\_n\}$, denoted as $T\_n$.

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