
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\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\\), \\(S_{11}=11b_4\).  
(Ⅰ) Find the general formula for \\(\{a_n\}\) and \\(\{b_n\}\);  
(Ⅱ) Calculate the sum of the first \\(n\\) terms of the sequence \\(\{a_{2n}b_n\}\) \\((n\in\mathbb{N}^*)\).

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