
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, including those with any orientation and side lengths of one or two units, have all vertices on this lattice? Assume three of the six outer vertices are offset by 120 degrees relative to the other three. [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy]

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