
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 the geometric sequence $\{a_n\}$ is an increasing sequence, $a_2a_5=32$, $a_3+a_4=12$, and the sequence $\{b_n\}$ satisfies $b_n=2\log_2a_{n+1}$, $S_n$ is the sum of the first $n$ terms of the sequence $\{b_n\}$  
(1) Find $S_n$;  
(2) If for any $n\in\mathbb{N}_+$, the inequality $\frac{S_n}{a_n}\leqslant \frac{S_k}{a_k}$ holds, find the value of the positive integer $k$.

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