
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
In an equilateral triangle $ABC$, 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 (triangle $FGH$) to the non-shaded area in triangle $ABC$. 

[asy]
draw((0,0)--(1,0)--(0.5,sqrt(3)/2)--cycle,linewidth(1));
filldraw((0,0)--(0.5,0)--(0.5,sqrt(3)/4)--(0,sqrt(3)/2)--cycle,gray, linewidth(1));
filldraw((0.5,0)--(0.5,sqrt(3)/8)--(0.25,sqrt(3)/2)--(0,sqrt(3)/2)--cycle,white,linewidth(1));

label("A", (0.5,sqrt(3)/2), N);
label("B", (0,0), SW);
label("C", (1,0), SE);
label("D", (0.75,sqrt(3)/4), NE);
label("E", (0.5,0), S);
label("F", (0.25,sqrt(3)/4), NW);
label("G", (0.5,sqrt(3)/4), N);
label("H", (0.5,sqrt(3)/8), E);
[/asy]

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