
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 triangle \( ABC \), \( AC = 3 AB \). Let \( AD \) bisect angle \( A \) with \( D \) lying on \( BC \), and let \( E \) be the foot of the perpendicular from \( C \) to \( AD \). Find \( \frac{[ABD]}{[CDE]} \). (Here, \([XYZ]\) denotes the area of triangle \( XYZ \)).

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