
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 $f(x)=\sin x\cos x-\sqrt{3}\sin^2 x+\frac{\sqrt{3}}{2}$.  
$(1)$ Find the interval where the function $f(x)$ is monotonically increasing.  
$(2)$ If $f(x_{0})=\frac{3}{5}$ and $x_{0}\in[\frac{\pi}{6},\frac{\pi}{3}]$, find the value of $\cos 2x_{0}$.

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