
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
Determine the largest \( N \) for which there exists a table \( T \) of integers with \( N \) rows and 100 columns that has the following properties:
(i) Every row contains the numbers \( 1, 2, \ldots, 100 \) in some order.
(ii) For any two distinct rows \( r \) and \( s \), there is a column \( c \) such that \( |T(r, c) - T(s, c)| \geq 2 \). Here \( T(r, c) \) means the number at the intersection of the row \( r \) and the column \( c \).

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