
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 an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit between adjacent points.

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