
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
The graphs of the equations
\[ y=k, \quad y=\sqrt{3}x+3k, \quad y=-\sqrt{3}x+3k, \]
are drawn in the coordinate plane for \( k=-8, -7, -6, \ldots, 7, 8. \) These 51 lines cut part of the plane into equilateral triangles of side \( 3/\sqrt{3}. \) How many such triangles are formed?

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