
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 $m > 0$, point $A(4,m)$ is on the parabola $y^{2}=2px (p > 0)$. $F$ is the focus, and circle $C$ with center $A$ and radius $|AF|$ intercepts a chord of length $6$ on the $y$-axis. What is the standard equation of circle $C$?

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