
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
Consider the parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$. A line with slope $\sqrt {3}$ intersects $C$ at point $A$ (above the $x$-axis), and point $B$ lies on the axis of symmetry $l$ of $C$. If $AB$ is perpendicular to $l$ and $|FB|=4$:

(I) Find the equation of the parabola $C$.

(II) Given a point $P$ on the line $m$: $x=2$, two distinct lines with slopes $k_{1}$ and $k_{2}$ are drawn through $P$, each intersecting parabola $C$ at exactly one point. If $k_{1}=2k_{2}$, find the coordinates of point $P$.

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