
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: $f(\alpha)=\frac{2\cos^{2}\alpha-\sin2\alpha}{2(\cos\alpha-\sin\alpha)}$.  
$(1)$ Simplify $f(\alpha)$;  
$(2)$ If $\alpha$ is an angle in the second quadrant and $\sin\alpha=\frac{3}{5}$, find $f(\alpha+\frac{\pi}{6})$.

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