
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
A hexagon $ABCDEF$ has its sides doubled in length compared to a regular hexagon, resulting in it being split into 24 smaller equilateral triangles. When every third vertex (from a larger, subdivided hexagon) is connected, a larger equilateral triangle, $\triangle ACF$, is formed. Compute the ratio $[\triangle ABM]/[\triangle ACF]$, where $M$ is the midpoint of segment $AB$. [asy]
size(200); defaultpen(linewidth(0.8)); dotfactor=5;
pair[] hex = new pair[6];
string[] hexlabels = {"$C$","$B$","$A$","$F$","$E$","$D$"};
hexlabels.cyclic=true;
hex[0] = dir(0);
for(int i = 1; i <= 6; ++i){
hex[i] = 2*dir(60*i); //doubling the length
draw(hex[i] -- hex[i-1]);
dot(hexlabels[i],hex[i],hex[i]);
}
pair M = (hex[2] + hex[1])/2;
dot("$M$", M, dir(75));
draw(hex[0]--hex[2]--hex[4]--cycle,linewidth(1.3));
draw(hex[1]--hex[2]--M--cycle,linewidth(1.3));
[/asy]

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