
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
Triangle $ABC$ is equilateral with side length $AB = 2$. Points $E$ and $G$ are on line $\overline{AB}$, and points $D$ and $F$ are on line $\overline{AC}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have equal areas. Determine the total length of $DE + FG$.
A) $\frac{2}{\sqrt{3}}$
B) $\frac{3}{\sqrt{3}}$
C) $\frac{4}{\sqrt{3}}$
D) $\frac{5}{\sqrt{3}}$

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