
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 of positive terms $\{a_{n}\}$ that satisfies $\log _{2}a_{n+2}+(-1)^{n}\log _{2}a_{n}=1$, with $a_{1}=1$ and $a_{2}=2$.<br/>$(1)$ Given $b_{n}=a_{2n-1}$, find the general formula for $\{b_{n}\}$;<br/>$(2)$ Find the sum of the first $2023$ terms of the sequence $\{a_{n}\}$ denoted as $S_{2023}$.

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