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

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