
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
In a geometric sequence $\\{a\_n\\}$, $a\_n > 0 (n \in \mathbb{N}^*)$, the common ratio $q \in (0, 1)$, and $a\_1a\_5 + 2aa\_5 + a\_2a\_8 = 25$, and the geometric mean of $a\_3$ and $a\_5$ is $2$.

(1) Find the general term formula of the sequence $\\{a\_n\\}$;

(2) If $b\_n = \log_2 a\_n$, find the sum of the first $n$ terms $S\_n$ of the sequence $\\{b\_n\\}$;

(3) Determine whether there exists $k \in \mathbb{N}^*$ such that $\frac{S\_1}{1} + \frac{S\_2}{2} + \dots + \frac{S\_n}{n} < k$ holds for any $n \in \mathbb{N}^*$. If so, find the minimum value of $k$; if not, explain the reason.

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