
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 an arithmetic sequence $\{ a\_n \}$ with the sum of its first $n$ terms denoted as $S\_n$. The sequence $S\_3$, $\frac{S\_5}{2}$, $S\_4$ forms an arithmetic sequence, and $a\_5 = 3a\_2 + 2a\_1 - 2$.

1. Find the general term formula for the sequence $\{ a\_n \}$.
2. Let $b\_n = 2^{n-1}$. Find the sum $T\_n$ of the first $n$ terms of the sequence $\{ \frac{a\_n}{b\_n} \}$.

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