
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 plane rectangular coordinate system $xOy$, given that the focus of the parabola $C:y^{2}=2px\left(p \gt 0\right)$ is $F({\sqrt{p},0})$, then the equation of $C$ is ______; if points $P$ and $F$ are symmetric about the $y$-axis, and a circle with diameter $PF$ intersects $C$ at point $A$, then $\cos \angle OAF=\_\_\_\_\_\_$.

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