
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 the geometric sequence $\{a_n\}$, $a_n > 0$ for $n \in \mathbb{N}^*$, and it's given that $a_1a_3 = 4$, and $a_3 + 1$ is the arithmetic mean of $a_2$ and $a_4$.

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

(II) Given the sequence $\{b_n\}$ satisfies $b_n = a_{n+1} + \log_2 a_n$ for $n = 1, 2, 3, \ldots$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$.

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