
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
Let  $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$  be a set consisting of points on the coordinate plane. Each element of  $S$  is colored one of four given colors. A subset  $T$  of  $S$  is called *colorful* if  $T$  consists of exactly  $4$  points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets  $S$  can have, among all legitimate coloring patters.

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