
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 an octagonal lattice is one unit from its nearest neighbor. Determine how many equilateral triangles, with all vertices on lattice points, exist within this lattice formed by 8 points surrounding a central point.

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