
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, given the parabola $\Gamma: y = a x^{2}$ (where $a$ is a positive real number). For a given positive real number $b$, if a circle with center $P(0, b)$ and variable radius $r(r > 0)$ intersects the parabola $\Gamma$ at four points $A, B, C,$ and $D$, find the maximum area $S$ of the quadrilateral with vertices $A, B, C,$ and $D$.

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