
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 \( n \) be a positive integer. An \( n \times n \) matrix (a rectangular array of numbers with \( n \) rows and \( n \) columns) is said to be a platinum matrix if:

(i) The \( n^2 \) entries are integers from 1 to \( n \);

(ii) Each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from 1 to \( n \) exactly once; and

(iii) There exists a collection of \( n \) entries containing each of the numbers from 1 to \( n \), such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix.

Determine all values of \( n \) for which there exists an \( n \times n \) platinum matrix.

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