
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 in the geometric sequence $\{a_n\}$, it is always true that $a_{n+1} > a_n$ for $n\in\mathbb{N}^*$, and $a_1a_4=8$, $a_2+a_3=6$.

(I) Find the general formula for the sequence $\{a_n\}$.

(II) If the sequence $\{b_n\}$ satisfies $\dfrac{a_1}{b_1} + \dfrac{3a_2}{b_2} + \ldots + \dfrac{(2n-1)a_n}{b_n} = n$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$.

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