
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 arithmetic sequence $\{a_n\}$, $a_2=5$, $a_1+a_4=12$. In the geometric sequence $\{b_n\}$, all terms are positive, and it satisfies $b_nb_{n+1}=2^{a_n}$.  
(I) Find the general formula for the sequence $\{a_n\}$ and the common ratio $q$ of the sequence $\{b_n\}$.  
(II) Find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n+b_n\}$.

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