
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
$y = k, \qquad y = \sqrt{3}x + k, \qquad y = -\sqrt{3}x + k,$
are drawn in the coordinate plane for $k = -5, -4, ..., 4, 5.\,$ These 33 lines cut part of the plane into equilateral triangles of side $1/\sqrt{3}.\,$ How many such triangles are formed?

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