
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 a set of points on the coordinate plane 
$$
S=\{(x, y) \mid x=1,2, \cdots, 1993, y=1,2,3,4\} .
$$

where \(T \subset S\) and any 4 points in \(T\) are not the vertices of a square, find the maximum value of \(|T|\).

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