
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
Given parallelogram $ABCD$ with coordinates $A(0, 0)$, $B(1, 2)$, $C(3, 2)$, and $D(2, 0)$. Let $E$ be the midpoint of diagonal $BD$ and $F$ a point on $DA$ such that $DF=\frac{1}{4}DA$. If $G$ is the midpoint of side $BC$, find the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABFG$.
A) $\frac{1}{4\sqrt{5} + 6}$  
B) $\frac{1}{8}$  
C) $\frac{1}{6}$  
D) $\frac{1}{2\sqrt{5} + 3}$  
E) $\frac{1}{4}$

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