
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 circle $C$ satisfy: $(1)$ The chord length obtained by intersecting the $y$-axis is $2$; $(2)$ It is divided into upper and lower arcs by the $x$-axis, with the length ratio of the arcs being $3:1$; $(3)$ The distance from the center of the circle to the line $l: x-2y=0$ is $d$. When $d$ is minimized, the area of circle $C$ is ______.

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