
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 that $f(\alpha) = \cos\alpha \sqrt{\frac{\cot\alpha - \cos\alpha}{\cot\alpha + \cos\alpha}} + \sin\alpha \sqrt{\frac{\tan\alpha - \sin\alpha}{\tan\alpha + \sin\alpha}}$, and $\alpha$ is an angle in the second quadrant.  
(1) Simplify $f(\alpha)$.  
(2) If $f(-\alpha) = \frac{1}{5}$, find the value of $\frac{1}{\tan\alpha} - \frac{1}{\cot\alpha}$.

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