
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

The center of ellipse \( \Gamma \) is at the origin \( O \). The foci of the ellipse lie on the x-axis, and the eccentricity \( e = \sqrt{\frac{2}{3}} \). A line \( l \) intersects the ellipse \( \Gamma \) at two points \( A \) and \( B \) such that \( \overrightarrow{C A} = 2 \overrightarrow{B C} \), where \( C \) is a fixed point \((-1, 0)\). When the area of the triangle \( \triangle O A B \) is maximized, find the equation of the ellipse \( \Gamma \).

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