
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 sequence $\{a\_n\}$ is a geometric sequence with all positive terms, and $a\_1=1$, $a\_2 \cdot a\_4=16$.

(I) Let $b\_n=\log\_2 a\_n$, find the general term formula of the sequence $\{b\_n\}$;

(II) Find the sum $S\_n$ of the first $n$ terms of the sequence $\{a\_n \cdot b\_n\}$.

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