
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 \(ABC\), \(AB=12\), \(BC=16\), 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 \(p^2+q^2+r^2=3/8\). 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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