
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 $0 < \alpha < \frac{\pi}{2} < \beta < \pi$, $\vec{a}=(\cos \alpha,3)$ and $\vec{b}=(-4,\sin \alpha)$, and $\vec{a} \perp \vec{b}$, $\cos (\beta-\alpha)= \frac{ \sqrt{2}}{10}$.

$(I)$ Find the values of $\tan \alpha$ and $\sin \alpha$;    $(II)$ Find the value of $\sin \beta$.

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