
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. Consider a larger hexagonal lattice that includes an additional layer of points making two concentric hexagons. How many equilateral triangles, including at least one vertex from the outer hexagon, have all three vertices in the lattice?

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