
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
Each point in the hexagonal lattice shown is one unit from its nearest neighbor. A second larger hexagon, concentric with the first, has been added to the original single hexagon, with each of its points two units from the nearest neighbor in the original hexagon. How many equilateral triangles, with all vertices either in the original hexagon, the second hexagon, or both, are there?
[asy]
size(75);
dot(origin); // Center Point
for(real x = 0; x < 360; x+=60){
    dot(dir(x)); // First Hexagon
    dot(2*dir(x)); // Second, larger Hexagon
}
[/asy]

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