
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 extended hexagonal lattice shown is one unit from its nearest neighbor. The lattice now includes an additional ring of points around the original hexagon. How many equilateral triangles, with all three vertices in the lattice, can be formed? 

[asy]
size(100);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
dot(2*dir(0));
dot(2*dir(30));
dot(2*dir(60));
dot(2*dir(90));
dot(2*dir(120));
dot(2*dir(150));
dot(2*dir(180));
dot(2*dir(210));
dot(2*dir(240));
dot(2*dir(270));
dot(2*dir(300));
dot(2*dir(330));
[/asy]

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