
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 the function $f(x)={\cos^2}x+\sqrt{3}\sin{x}\cos{x}+1$, $x\in \mathbb{R}$.  
$(1)$ Find the intervals where the function $y=f(x)$ is monotonically increasing.  
$(2)$ Find the range of the function $y=f(x)$ when $x\in \left[0,\frac{\pi}{2}\right]$.

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