
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, the distance from the center to the line $l$: $y=\tan \alpha \cdot x$ is $\sqrt{5}$, and the length of the chord intercepted by the line $l$ on the circle is $2\sqrt{2}$, find the equation of circle $C$.

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