
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, the sum of the first $n$ terms is $s_n (n \in N^*),$ and $\{b_n\}$ is a geometric sequence with the first term being $2$ and the common ratio greater than $0$, given 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 $\{b_n\}$.

$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_{2n}b_{2n-1}\}$.

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