
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 $f(x)=\cos^2x- \sqrt{3}\sin x\cos x+1$.  
(1) Find the interval of monotonic increase for the function $f(x)$.  
(2) If $f(\theta)= \frac{5}{6}$, where $\theta\in\left( \frac{\pi}{3}, \frac{2\pi}{3}\right)$, find the value of $\sin 2\theta$.

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