
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 $f'(x)$ is the derivative of the function $f(x)$, and both have a domain of $R$. If the functions $f'(\frac{x}{2}+1)$ and $f(x)-x$ are both even functions, write down a symmetric center for the function $\frac{f(x)}{x}$ as ______; $\left[{f'}(1)-1\right]\left[{f'}(2)+1\right]+\left[{f'}(2)-1\right]\left[{f'}(3)+1\right]+\left[{f'}(3)-1\right]\left[{f'}(4)+1\right]+\ldots +\left[{f'}(2023)-1\right]\left[{f'}(2024)+1\right]=\_\_\_\_\_\_$.

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