
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

On each side of a square, 100 points are selected. From each selected point, a line segment is drawn perpendicular to the corresponding side of the square into the interior of the square. It turns out that no two of these segments lie on the same line. Mark all the points of intersection of these segments. What is the largest value of \( k < 200 \) for which it may happen that exactly \( k \) marked points lie on each drawn segment?

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