
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 $y=k$, $y=x + 3k$, and $y=-x + 3k$ are plotted in the coordinate plane for $k=-6,-5,-4,\ldots,5,6$. These 39 lines divide part of the plane into equilateral triangles of side $1$. How many such triangles are formed?

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