
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
The sequence $\{a\_n\}$ has a sum of its first $n$ terms denoted as $S\_n$ and satisfies $S\_n = 2a\_n - 1$, $n \in \mathbb{N}^*$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. Let $T\_n = a\_1a\_2 - a\_2a\_3 + a\_3a\_4 - a\_4a\_5 + \dots + (-1)^{n+1}a\_n a\_{n+1}$. Find the general formula for the sequence $\{T\_n\}$.
3. Suppose there is an $m$-term sequence $\{b\_n\}$ of consecutive positive integers that satisfies: $\log\_2 2 + \log\_2 (1 + \frac{1}{b\_1}) + \log\_2 (1 + \frac{1}{b\_2}) + \dots + \log\_2 (1 + \frac{1}{b\_m}) = \log\_2 (\log\_2 a\_m)$. What is the maximum number of terms in the sequence $\{b\_n\}$? And find the sum of these terms.

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