
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 $ \vec{a}=(\sin 2x,\cos 2x) $ and $ \vec{b}=( \frac{1}{2}, \frac{ \sqrt{3}}{2})$, where $x\in\mathbb{R}$, and $f(x)= \vec{a}\cdot \vec{b}+|\vec{a}|+|\vec{b}|$.

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

$(2)$ If $x\in\left[ \frac{\pi}{6}, \frac{2\pi}{3}\right] $, find the maximum and minimum values of the function $f(x)$.

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