
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:
[list=i]
[*] the  $n^2$  entries are integers from  $1$  to  $n$ ;
[*] 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
[*] 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.
[/list]
Determine all values of  $n$  for which there exists an  $n \times n$  platinum matrix.

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