
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
Let the circle satisfy the following conditions: (1) The chord length intercepted by the y-axis is 2; (2) The circle is divided by the x-axis into two arc segments with a length ratio of 3:1. Among all circles that meet conditions (1) and (2), find the equation of the circle with the minimum distance from its center to the line \( l: x - 2y = 0 \).

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