
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 a sequence $\{a_{n}\}$ where $a_{1}=2$, and ${a_{n+1}}=\left\{\begin{array}{l}\frac{3}{2}{a_n}$, for odd $n$\\ $2{a_n}$, for even $n\end{array}\right.$.<br/>$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$;<br/>$(2)$ Find the sum of the first $12$ terms of the sequence $\{a_{n}\}$.

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