
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 an equilateral triangle with each side of length $s$. Points $D$, $E$, and $F$ are still the midpoints of sides $AB$, $BC$, and $CA$ respectively. Point $G$ is now the centroid of triangle $ADF$ and point $H$ is the centroid of triangle $BFE$. Determine the ratio of the shaded area, bounded by segments $DG$, $GH$, and $HF$ to the non-shaded area in triangle $ABC$.

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