
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
Given \( 2k \) points (\( k \geqslant 3 \)) in a plane, with no three points collinear, every pair of points is connected by a line segment which is either red or blue. A triangle is called monochromatic if all its three sides are of the same color. Let \( S \) denote the number of monochromatic triangles. Find the minimum value of \( S \) over all possible colorings.

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