
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
For a table  $n \times 9$  ( $n$  rows and  $9$  columns), determine the maximum of  $n$  that we can write one number in the set  $\left\{ {1,2,...,9} \right\}$  in each cell such that these conditions are satisfied:

1. Each row contains enough  $9$  numbers of the set  $\left\{ {1,2,...,9} \right\}$ .

2. Any two rows are distinct.

3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

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