
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 \( \triangle ABC \), angle \( \triangle C \) is a right angle, \( AC = 3 \) and \( BC = 4 \). In triangle \( \triangle ABD \), angle \( \triangle A \) is a right angle, and \( AD = 12 \). Points \( C \) and \( D \) are on opposite sides of \( AB \). A line passing through point \( D \) and parallel to \( AC \) intersects \( CB \) at \( E \). Given that \( \frac{DE}{DB} = \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, calculate \( \frac{DE}{DB} \).

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