
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
In the coordinate plane, draw the following lines \( y = k \), \( y = \sqrt{3} x + 2k \), and \( y = -\sqrt{3} x + 2k \), where \( k = 0, \pm 1, \pm 2, \ldots, \pm 10 \). These 63 lines divide the plane into several equilateral triangles. Find the number of equilateral triangles with a side length of \( \frac{2}{\sqrt{3}} \).

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