
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
On a plane, there are $n$ points, with any three points not collinear, and a line segment is drawn between any two points. Each line segment is colored either red or blue. A triangle with all three edges of the same color is called a "monochromatic triangle". Let $S$ denote the number of monochromatic triangles.

(1) If $n=6$, find the minimum value of $S$ for all possible colorings.

(2) If $n=2k$ (where $k \geq 4$ is an integer), find the minimum value of $S$ for all possible colorings.

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