
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 $f(x)=3\sin{x}\cos{x}-\sqrt{3}\cos^{2}x+\frac{\sqrt{3}}{2}$.  
$(1)$ Find the interval where $y=f(x)$ is monotonically increasing.  
$(2)$ When $x\in[\frac{\pi}{6},\frac{3\pi}{4}]$, find the range of $y=f(x)$.

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