
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 that the function $f(x)$ and its derivative $f'(x)$ have domains of all real numbers, and let $g(x) = f'(x)$. If the graph of the function $f(x)$ is symmetric about the point $(3,0)$, $g(2x+\frac{3}{2})$ is an even function, and $g(1) = 2$, $g(3) = -3$, then find the sum of $g(k)$ for $k=1$ to $2024$.

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