
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 $\sqrt{3}\tan\alpha=-2\sin\alpha$, where $\alpha \in \left(\frac{\pi}{2}, \pi\right)$.  
$(1)$ Find $\cos \alpha$;  
$(2)$ Find the value of $\cos\left(\alpha-\frac{\pi}{4}\right)+\sin\left(\alpha+\frac{\pi}{4}\right)$.

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