
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 $f(\alpha) = \frac{\sin(\frac{\pi}{2} - \alpha) + \sin(-\pi - \alpha)}{3\cos(2\pi + \alpha) + \cos(\frac{3\pi}{2} - \alpha)} = 3$  
(1) Find the value of $\frac{\sin\alpha - 3\cos\alpha}{\sin\alpha + \cos\alpha}$;  
(2) If the center of circle C is on the x-axis, and the distance from the center to the line $l: y = \tan\alpha \cdot x$ is $\sqrt{5}$, and the chord intercepted by line $l$ on the circle has a length of $2\sqrt{2}$, find the equation of circle C.

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