
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 the terminal side of angle $\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \frac{\sqrt{15}}{4})$.  
$(1)$ Find the value of the real number $m$;  
$(2)$ Let $f(\alpha) = \frac{\cos(2\pi - \alpha) + \tan(3\pi + \alpha)}{\sin(\pi - \alpha) \cdot \cos(\alpha + \frac{3\pi}{2})}$. Find the value of $f(\alpha)$.

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