Please reason step by step, and put your final answer within \boxed{}.
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 $C$?
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