
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 $s$. 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 formed inside triangle $ABC$ to the non-shaded area in the same triangle. The triangle is shaded as follows: the ever smaller triangles formed within each preceding midpoint triangle are alternately shaded and non-shaded starting from near vertex A.

[asy]
size(6cm);
pair A = (0, sqrt(3)), B = (-1, 0), C = (1, 0);
pair D = midpoint(A--B), E = midpoint(B--C), F = midpoint(C--A);
pair G = midpoint(D--F), H = midpoint(F--E);
draw(A--B--C--cycle);
filldraw(D--E--F--cycle, gray);
filldraw(G--D--F--cycle, white);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, W);
label("$E$", E, S);
label("$F$", F, E);
label("$G$", G, W);
label("$H$", H, SE);
[/asy]

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