
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 \( k = -12, -11, -10, \ldots, 11, 12 \). These 75 lines partition 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}
user