
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 an arithmetic sequence $\{a_{n}\}$ that satisfies $a_{n}+a_{n+1}=4n$, where $n\in N^{*}$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let ${b}_{1}=1,{b}_{n+1}=\left\{\begin{array}{l}{a}_{n},n\text{ is odd}\\-{b}_{n}+{2}^{n},n\text{ is even}\end{array}\right.$, find the sum of the first $2n$ terms of the sequence $\{b_{n}\}$ denoted as $S_{2n}$.

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