
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
Given the lines in the coordinate plane:
\[ y = m, \quad y = \sqrt{3}x + 3m, \quad y = -\sqrt{3}x + 3m, \]
for \(m = -12, -11, \ldots, 11, 12\), which form 75 lines in total, determine how many equilateral triangles of side length \(3/\sqrt{3}\) are formed by these intersections within the formed hexagonal region.

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