
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 rectangle $ABCD$, side $DC = 3 \cdot CB$. Points $E$ and $F$ are defined on line segment $\overline{AB}$ in such a way that lines $\overline{ED}$ and $\overline{FD}$ quadrusect $\angle ADC$. Determine the ratio of the area of triangle $\triangle DEF$ to the area of rectangle $ABCD$.
A) $\frac{1}{12}$
B) $\frac{1}{12\sqrt{2}}$
C) $\frac{1}{6}$
D) $\frac{\sqrt{2}}{12}$
E) $\frac{\sqrt{3}}{16}$

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