
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 $\{a_{n}\}$ where $a_{1}=1$, and ${a}_{n+1}^{2}-{a}_{n}^{2}=8n$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let $b_{n}=a_{n}\cdot \sin (\frac{{a}_{n}}{2}\cdot \pi )$, calculate the sum of the first $2023$ terms of the sequence $\{b_{n}\}$.

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