
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 an arithmetic sequence $\{a_n\}$ satisfies $a_4=5$, $a_2+a_8=14$, and a sequence $\{b_n\}$ satisfies $b_1=1$, $b_{n+1}=2^{a_n+3} \cdot b_n$.
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Find the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{\log_2 b_{n+1}} \right\}$;
$(3)$ If $c_n=a_n \cdot (\sqrt{2})^{a_n+1}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, $S_n$.

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