
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  $j$ ,  ${1\leq i,j\leq n}$ . We say that  $A$  is *golden* if  ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$  are distinct sets. Find the least integer  $n$  such that there exists a  ${2004\times 2004}$  golden matrix with entries in the set  ${\{1,2,\dots ,n\}}$ .

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