
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 \(\Gamma: \frac{x^{2}}{4} + y^{2} = 1\) with two chords \(AB\) and \(CD\) such that \(\overrightarrow{DC} = 2 \overrightarrow{AB}\), and let the lines \(AD\) and \(BC\) intersect at point \(P\).

1. Find the maximum value of \(|AB| + |CD|\).
2. If point \(P\) lies on the parabola \(y = \frac{1}{4}x^{2} - 3\), find the maximum area of the quadrilateral \(ABCD\).

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