
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 the triangle \( \triangle ABC \), \( \angle C = 90^{\circ} \), and \( CB > CA \). Point \( D \) is on \( BC \) such that \( \angle CAD = 2 \angle DAB \). If \( \frac{AC}{AD} = \frac{2}{3} \) and \( \frac{CD}{BD} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, determine the value of \( \frac{m}{n} \).

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