
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 that satisfies the following conditions: (1) the length of the chord intercepted on the y-axis is 2; (2) the ratio of the arc lengths divided by the x-axis is 3:1; (3) the distance from the center of the circle to the line $l$: $x - 2y = 0$ is $\frac{\sqrt{5}}{5}$. Find the equation of the circle.

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