
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 $\vec{a}=(3\sin \alpha,\cos \alpha)$ and $\vec{b}=(2\sin \alpha,5\sin \alpha-4\cos \alpha)$, where $\alpha\in(\frac{3\pi}{2},2\pi)$, and $\vec{a} \perp \vec{b}$.

(I) Find the value of $\tan \alpha$;

(II) Find the value of $\cos (2\alpha+ \frac{\pi}{3})$.

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