
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. How many equilateral triangles have all three vertices in the lattice, where the lattice is extended to include points at a distance of 2 units from the center? [asy]size(100);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
dot(dir(0)*2);
dot(dir(60)*2);
dot(dir(120)*2);
dot(dir(180)*2);
dot(dir(240)*2);
dot(dir(300)*2);
[/asy]

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