
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
In the coordinate plane, a set of  $2000$  points  $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$  is called *good* if  $0\leq x_i \leq 83$ ,  $0\leq y_i \leq 83$  for  $i = 1, 2, \dots, 2000$  and  $x_i \not= x_j$  when  $i\not=j$ . Find the largest positive integer  $n$  such that, for any good set, the interior and boundary of some unit square contains exactly  $n$  of the points in the set on its interior or its boundary.

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