
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 \( ABC \), where \( AB = 12 \), \( BC = 16 \), and \( CA = 20 \). Point \( D \) is on \( \overline{AB} \), \( E \) is on \( \overline{BC} \), and \( F \) is on \( \overline{CA} \). Let \( AD = p \cdot AB \), \( BE = q \cdot BC \), and \( CF = r \cdot CA \), where \( p \), \( q \), and \( r \) are positive and satisfy \( p + q + r = 3/4 \) and \( p^2 + q^2 + r^2 = 1/2 \). The ratio of the area of triangle \( DEF \) to the area of triangle \( ABC \) can be written in the form \( m/n \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \).

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