
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 $\overrightarrow{m}=(\cos^2x,\sqrt{3})$ and $\overrightarrow{n}=(2,\sin 2x)$, and a function $f(x)=\overrightarrow{m} \cdot \overrightarrow{n}$.  
$(1)$ Find the interval where the function $y=f(x)$ is monotonically increasing.  
$(2)$ If $x \in [0,\frac{\pi}{2}]$, determine the range of the function $y=f(x)$.

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