
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 vectors $\overrightarrow{a}=(2,\sqrt{3})$ and $\overrightarrow{b}=(\frac{1}{2}-\cos^{2}\frac{x}{2},\sin x)$, and the function $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$.  
$(1)$ Find the interval of $x$ where $f(x)$ is monotonically increasing.  
$(2)$ If all the points on the graph of function $f(x)$ have their y-coordinates unchanged and the x-coordinates become half of the original, resulting in the function $g(x)$, and the equation $g(x)-m=0$ has a solution in the interval $[0,\frac{\pi}{2}]$, find the range of real number $m$.

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