
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 a triangle \( \triangle ABC \) and a point \( D \) on its side \( BC \) such that \( AB = 2BD \) and \( AC = 3CD \), and an ellipse \( \Gamma \) can be constructed with \( A \) and \( D \) as foci passing through points \( B \) and \( C \), find the eccentricity of \( \Gamma \).

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