
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 $\{a_{n}\}$ where $a_{1}=3$, and ${a_{n+1}}=\left\{{\begin{array}{l}{2{a_n},n \text{ is even,}}\\{{a_n}-1,n \text{ is odd.}}\end{array}}\right.$<br/>$(1)$ Let $b_{n}=a_{2n}+a_{2n-1}$, find the general formula for the sequence $\{b_{n}\}$;<br/>$(2)$ Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, find the smallest value of $n$ that satisfies the inequality $S_{n} \gt 2023$.

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