
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
Consider a larger hexagonal lattice where each point is one unit from its nearest neighbor and includes an additional layer of points around the original hexagon, making the furthest points two units from the center. How many equilateral triangles, having all vertices in this expanded lattice, can be formed?

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