
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 a sequence $\{a_n\}$ satisfying the following condition for $n \in \mathbb{N}^*$: 
$$a_{1}+ \frac {a_{2}}{2}+ \frac {a_{3}}{2^{2}}+\ldots+ \frac {a_{n+1}}{2^{n}}=2n+2,$$
with $a_2=4$.
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n=\log_{\sqrt{2}}a_n$. Calculate the sum of the first $n$ terms of the sequence $\{a_nb_n\}$, denoted by $S_n$.

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