
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 the set of points \( S = \{(x, y) \mid x=1,2, \cdots, 1993, y=1,2,3,4\} \) in the coordinate plane, where \( T \subset S \) and any 4 points in \( T \) are not the vertices of a square, find the maximum size of \( |T| \).

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