
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 on the displayed triangular lattice is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices are all on lattice points. [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
dot(dir(30), red);
dot(dir(90), red);
dot(dir(150), red);
dot(dir(210), red);
dot(dir(270), red);
dot(dir(330), red);
[/asy]

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