
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
Circle $C$ with its center in the first quadrant, intersects the $y$-axis with a chord length of $2$, and intersects the $x$-axis with a chord length of $4$. The ratio of the lengths of the two segments of the circle cut by the $x$-axis is $3:1$. Find the equation of circle $C$.

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