
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 ellipse \(\Gamma\):

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]

Let \(A\) and \(B\) be the left and top vertices of the ellipse \(\Gamma\), respectively. Let \(P\) be a point in the fourth quadrant on the ellipse \(\Gamma\). The line segment \(PA\) intersects the y-axis at point \(C\), and the line segment \(PB\) intersects the x-axis at point \(D\). Find the maximum area of the triangle \(\triangle PCD\).

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