
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 a sequence $\{a\_n\}$ where all terms are positive numbers, it satisfies the equation $\log\_2 a\_n = 1 + \log\_2 a\_{n-1}, n \in \mathbb{N}^*, n \geq 2$, and $a\_1 = 2$.
(I) Find the general formula for the sequence $\{a\_n\}$;
(II) Let $c\_n = (3n - 1) \cdot a\_n$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$;
(III) Suppose a sequence $\{b\_n\}$ satisfies $b\_n = \frac{n a\_n}{(2n + 1) \cdot 2^n}$, determine if there exist positive integers $m, n (1 < m < n)$ that make $b\_1$, $b\_m$, and $b\_n$ form a geometric sequence? If so, find all the values of $m$ and $n$; if not, explain the reason.

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