
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  $ABCD$  be a square with side  $4$ . Find, with proof, the biggest  $k$  such that no matter how we place  $k$  points into  $ABCD$ , such that they are on the interior but not on the sides, we always have a square with sidr length  $1$ , which is inside the square  $ABCD$ , such that it contains no points in its interior(they can be on the sides).

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