
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
Find the equation of the circle that satisfies the following conditions: (1) The chord it cuts on the y-axis is 2 units long; (2) The x-axis divides the circle into two arcs whose lengths are in the ratio 3:1. Among all such circles, find the one whose center is at the minimum distance from the line $l: x-2y=0$.

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