
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 a common ratio $q \neq 1$, the sum of the first $n$ terms is $S\_n$, $a\_1 + a\_3 = \frac{S\_4}{S\_2}$, and $(a\_1 - 1)$, $(a\_2 - 1)$, $(a\_3 - 1)$ are the first, second, and fifth terms of an arithmetic sequence, respectively.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n = a\_n \log\_2 a\_n$. Find the sum of the first $n$ terms, denoted as $T\_n$, for the sequence $\{b\_n\}$.

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