
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 side length 6. Points $D$, $E$, and $F$ are the midpoints of sides $AB$, $BC$, and $CA$, respectively. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. Calculate the ratio of the shaded area to the non-shaded area in triangle $ABC$. Assume the segments $DE$, $EF$, and $FD$ also partition the triangle into shaded and non-shaded regions.

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