
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 $\overline{a}=(\cos \frac{3\theta}{2},\sin \frac{3\theta}{2})$ and $\overline{b}=(\cos \frac{\theta}{2},-\sin \frac{\theta}{2})$, where $\theta\in[0, \frac{\pi}{3}]$,
(I) Find the maximum and minimum values of $\frac{\overline{a} \cdot \overline{b}}{|\overline{a}+\overline{b}|}$;
(II) If $|k\overline{a}+\overline{b}|=\sqrt{3}|\overline{a}-k\overline{b}| (k\in\mathbb{R})$, find the range of $k$.

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