
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 measuring $s$. 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 $ABC$, where the shaded regions are those within the inner triangle $DEF$ and additional trapezoids connecting $DEF$ with vertices $A$, $B$, and $C$.

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