
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 the parabola $C: y^{2}=2px\left(p \gt 0\right)$ and the circle $E: \left(x-4\right)^{2}+y^{2}=12$ have exactly two common points.<br/>$(1)$ Find the equation of the parabola $C$;<br/>$(2)$ Let $O$ be the origin, a line passing through the center of the circle $E$ intersects the circle $E$ at points $A$ and $B$, and the lines $OA$ and $OB$ intersect the parabola $C$ at points $P$ and $Q$ (points $P$ and $Q$ are not coincident with point $O$). Let the area of $\triangle OAB$ be $S_{1}$ and the area of $\triangle OPQ$ be $S_{2}$. Find the maximum value of $\frac{S_{1}}{S_{2}}$.

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