
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 a circle C with its center on the line $x-3y=0$ and tangent to the y-axis at the point (0,1).
(I) Find the equation of circle C;
(II) If circle C intersects the line $l: x-y+m=0$ at points A and B, and the lines connecting the center C with points A and B are perpendicular to each other ($CA \perp CB$), find the value of m.

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