
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 cells of a \(100 \times 100\) square, the numbers \(1, 2, \ldots, 10000\) are placed, each one appearing exactly once. Additionally, numbers that differ by 1 are written in adjacent cells sharing a side. After this, the distances between the centers of any two cells with numbers differing by exactly 5000 are counted. Let \(S\) be the minimum of these distances. What is the maximum possible value of \(S\)?

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