
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
The sequence $\{a\_n\}$ satisfies: $a\_1+ \frac{a\_2}{2}+ \frac{a\_3}{2^{2}}+...+ \frac{a_{n+1}}{2^{n}}=2n+2\ (n\in\mathbb{N}^{\ast})$, and $a\_2=4$.
(1) Find the general term formula for the sequence $\{a\_n\}$;
(2) If $b\_n=\log_{\sqrt{2}}a\_n$, find the sum $S\_n$ of the first $n$ terms of the sequence $\{a\_n b\_n\}$.

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