
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 integer  $N$ , for which there exists a  $6\times N$  table  $T$  that has the following properties: $*$  Every column contains the numbers  $1,2,\ldots,6$  in some ordering. $*$  For any two columns  $i\ne j$ , there exists a row  $r$  such that  $T(r,i)= T(r,j)$ . $*$  For any two columns  $i\ne j$ , there exists a row  $s$  such that  $T(s,i)\ne T(s,j)$ .

(Proposed by Gerhard Woeginger, Austria)

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