
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 $f(x)=(\sin x+\sqrt{3}\cos x)(\cos x-\sqrt{3}\sin x)$.

    $(1)$ Find the interval of monotonic increase for the function $f(x)$;

    $(2)$ If $f(x_{0})=\frac{6}{5}$, $x_{0}\in\left[0, \frac{\pi}{2}\right]$, find the value of $\cos 2x_{0}$.

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