
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 a circle $C$ with its center on the line $x-3y=0$ is tangent to the negative half-axis of the $y$-axis, and the chord cut by $C$ on the $x$-axis is $4\sqrt{2}$ in length, then the equation of circle $C$ is ____.

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