
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 all terms of the sequence $\{a\_n\}$ are positive numbers, the sum of the first $n$ terms is $S\_n$, and $2a\_1a\_n = S\_1 + S\_n$ holds for all positive integers $n$.

(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) Let $b\_n = \log\_2 a\_{n+1}$, find the sum of the first $n$ terms, $T\_n$, for the sequence $\{\frac{b\_n}{a\_n}\}$.

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