
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 the geometric sequence $\{a_n\}$ with a common ratio $q \in (0,1)$, it is given that $a_3+a_5=5$ and $a_2a_6=4$. Let $b_n = \log_2 a_n$ and let the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$. Find the value of $n$ for which the expression $\frac{S_1}{1} + \frac{S_2}{2} + \ldots + \frac{S_n}{n}$ reaches its maximum value.

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