
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 $\{a_n\}$ that satisfies $a_n=2+2\cos^2 \frac{n\pi}{2}$, where $n\in\mathbb{N}^*$, and an arithmetic sequence $\{b_n\}$ that satisfies $a_1=2b_1$, $a_2=b_2$.  
$(1)$ Find $b_n$;  
$(2)$ Let $c_n=a_{2n-1}b_{2n-1}+a_{2n}b_{2n}$, find $c_n$;  
$(3)$ Find the sum of the first $2n$ terms of the sequence $\{a_nb_n\}$, denoted as $S_{2n}$.

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