
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
A sequence $\{a\_n\}$ satisfies the equation $S\_n = 2a\_n - a\_1$, where $S\_n$ represents the sum of the first $n$ terms of the sequence. Additionally, $a\_1$, $a\_3 + 1$, and $a\_4$ form an arithmetic sequence. Let $b\_n = \log\_2(a\_1) + \log\_2(a\_2) + \dots + \log\_2(a\_n)$.

(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) Define a new sequence $\{c\_n\} = \{\frac{1}{b\_n} + (-1)^n a\_n\}$. Find the sum of the first $n$ terms of this sequence, denoted by $T\_n$.

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