
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 $y^{2}=4x$ and the circle $E:(x-4)^{2}+y^{2}=12$. Let $O$ be the origin. The line passing through the center of circle $E$ intersects the circle at points $A$ and $B$. The lines $OA$ and $OB$ intersect the parabola at points $P$ and $Q$ (points $P$ and $Q$ are not coincident with point $O$). Let $S_{1}$ be the area of triangle $OAB$ and $S_{2}$ be the area of triangle $OPQ$. Find the maximum value of $\frac{S_{1}}{S_{2}}$.

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