
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 that the center of circle $C$ lies on the graph of the function $y=x^{2}$, and the length of the line segment intercepted by circle $C$ on the $y$-axis is $8$. If circle $C$ passes through the origin, find the equation of circle $C$.

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