
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
Regular hexagon $ABCDEF$ is divided into twelve smaller equilateral triangles by adding a point $G$ in the center, and connecting it to each vertex and the midpoint of each side, as shown below. By connecting vertices $A$, $D$, and $F$, we obtain another larger equilateral triangle $\triangle ADF$. Compute the ratio $[\triangle ABG]/[\triangle ADF]$, where $G$ is the central point in the hexagon. [asy]
size(150); defaultpen(linewidth(0.8)); dotfactor=5;
pair[] hex = new pair[6];
pair G=(0,0);  // central point
string[] hexlabels = {"$C$","$B$","$A$","$F$","$E$","$D$"};
string Glabel="$G$";
hexlabels.cyclic=true;
hex[0] = dir(0);
for(int i = 1; i <= 6; ++i){
  hex[i] = dir(60*i);
  draw(hex[i] -- hex[i-1]);
  draw(G -- hex[i]);  // connect G to each vertex
  dot(hexlabels[i],hex[i],hex[i]);
}
draw(G--hex[0]); draw(G--hex[3]); // connect G to A and D
draw(hex[0]--hex[2]--hex[4]--cycle,linewidth(1.3)); // larger triangle ADF
draw(G--hex[1]--hex[2]--cycle,linewidth(1.3)); // ABG
dot(Glabel,G,G);
[/asy]

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