
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 positive integer \( N \) such that there exists a \( 6 \times N \) table \( T \) satisfying:
(1) Each column is a permutation of \( 1, 2, \cdots, 6 \);
(2) For any two columns \( i \neq j \), there exists a row \( r \) (where \( r \in \{1, 2, \cdots, 6\} \)) such that \( t_{ri} = t_{rj} \);
(3) For any two columns \( i \neq j \), there exists a row \( s \) (where \( s \in \{1, 2, \cdots, 6\} \)) such that \( t_{si} \neq t_{sj} \).

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