
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 table  $T$  of integers with  $N$  rows and  $100$  columns that has the following properties: $\text{(i)}$  Every row contains the numbers  $1$ ,  $2$ ,  $\ldots$ ,  $100$  in some order. $\text{(ii)}$  For any two distinct rows  $r$  and  $s$ , there is a column  $c$  such that  $|T(r,c) - T(s, c)|\geq 2$ . (Here  $T(r,c)$  is the entry in row  $r$  and column  $c$ .)

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