
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 that $\{a_{n}\}$ is an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}(n\in N^{*})$, $\{b_{n}\}$ is a geometric sequence with the first term being $2$ and a common ratio greater than $0$. It is known that $b_{2}+b_{3}=12$, $b_{3}=a_{4}-2a_{1}$, and $S_{11}=11b_{4}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Determine the sum of the first $n$ terms of the sequence $\{a_{2n}b_{n}\}$, where $n\in N^{*}$.

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