
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
Consider an arithmetic sequence $\{a\_n\}$. It is given that $a\_2+a\_3+a\_4=15$ and $a\_5=9$.
(I) Find the general term formula of the sequence $\{a\_n\}$.
(II) Let $b\_n=3^{\frac{a\_n+1}{2}}$. Find the sum $S\_n$ of the first $n$ terms of the sequence $\{\frac{a\_n+1}{2}\cdot b\_n\}$.

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