
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 and the sum of its first $n$ terms is $S_n$ $(n\in\mathbb{N^{+}})$, $\{b_n\}$ is a geometric sequence with the first term being 2 and the common ratio greater than 0, $b_2 + b_3 = 12$, $b_3 = a_4 - 2a_1$, and $S_{11} = 11b_4$.

(I) Find the general term formulas for $\{a_n\}$ and $\{b_n\}$.
(II) Find the sum of the first $n$ terms of the sequence $\{a_{2n}b_{2n-1}\}$, denoted as $T_n$ $(n\in\mathbb{N^{+}})$.

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