
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
Let \( A \) be an \( m \times m \) matrix. Let \( X_{i} \) be the set of entries in the \( i \)-th row and \( Y_{j} \) the set of entries in the \( j \)-th column, \( 1 \leq i, j \leq m \). \( A \) is called cool if the sets \( X_{1}, \ldots, X_{m}, Y_{1}, \ldots, Y_{m} \) are all different. Determine the smallest value for \( n \) so that a cool \( 2005 \times 2005 \) matrix with entries from the set \( \{1,2, \ldots, n\} \) exists.

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