
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
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and ${S}_{n}+(-1)^{n}{a}_{n+1}={3}^{n}$, $b_{n}=a_{2n+1}+2a_{2n}$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ If $a_{1}$, $a_{2}$, $a_{3}$ form an arithmetic sequence, find $S_{2n-1}$.

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