
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 $\vec{a}=(4,5\cos \alpha)$ and $\vec{b}=(3,-4\tan \alpha)$, where $\alpha \in (0,\frac{\pi}{2})$ and $\vec{a} \bot \vec{b}$.

$(1)$ Find $|\vec{a}-\vec{b}|$;  
$(2)$ Find $\sin \left( \frac{3\pi}{2}+2\alpha \right)+\cos \left( 2\alpha -\pi \right)$.

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