
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
In the Cartesian coordinate system $xOy$, the parabola $y^2=4x$ has its focus at $F$. Point $P$ is on the parabola and is above the $x$-axis. If the distance from point $P$ to the origin $O$ is $4\sqrt{2}$, then the equation of the circle passing through points $F$, $O$, and $P$ is \_\_\_\_\_\_.

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