
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 $\overrightarrow{a}=(\sqrt{3}\sin x,2\cos^{2}x)$, $\overrightarrow{b}=(2\cos x,1)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(1)$ Find the coordinates of the center of symmetry of the graph of $f(x)$ and the interval of monotonic increase for $x\in[0,\frac{\pi}{2}]$.  
$(2)$ If the function $g(x)=f(\frac{\pi x}{4})$, calculate the value of $g(1)+g(2)+g(3)+\cdots +g(2023)$.

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