
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 positive integers \( m \) and \( n \), if each row of an \( m \times n \) table is a permutation of \( 1, 2, \ldots, n \), and each column, after an appropriate rearrangement, forms an arithmetic sequence of length \( m \) with a common difference greater than 0, then this table is called an \( m \times n \) continuous arithmetic table.

Find all positive integers \( n \) such that a \( 3 \times n \) continuous arithmetic table exists.

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