
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 linear function $f(x)$ that satisfies $f(f(x))=x+3$.  
$(1)$ Find the analytical expression of $f(x)$.  
$(2)$ If $g(x)=\frac{x}{f(x)-\frac{1}{2}}$, find the value of $g(1)+g(2)+⋯+g(2023)+g(\frac{1}{2023})+g(\frac{1}{2022})+⋯+g(\frac{1}{2})$.

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