
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 vectors $\overrightarrow{a}=(\cos x,-\sqrt {3}\cos x)$ and $\overrightarrow{b}=(\cos x,\sin x)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.
(I) Find the interval where the function $f(x)$ is monotonically increasing.
(II) If $f(\theta)=\frac{5}{6}$, $\theta\in(\frac{\pi}{3},\frac{2\pi}{3})$, find the value of $\sin 2\theta$.

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