
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 an \(n \times n\) matrix \(A\), let \(X_{i}\) be the set of entries in row \(i\), and \(Y_{j}\) the set of entries in column \(j\), where \(1 \leq i, j \leq n\). We say that \(A\) is golden if \(X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{n}\) are distinct sets. Find the least integer \(n\) such that there exists a \(2004 \times 2004\) golden matrix with entries in the set \(\{1, 2, \ldots, n\}\).

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