
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 \( S=\{(i, j) \mid i, j=1, 2, \ldots, 100\} \) as the set of \( 100 \times 100 \) integer points on a Cartesian plane. Each point in \( S \) is colored with one of four given colors. Determine the maximum possible number of rectangles, with vertices points from \( S \) and sides parallel to the coordinate axes, whose four vertices each have a different color.

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