
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.  The numbers  $\{1, 2, ..., n^2\}$  are placed in an  $n \times n$  grid, each exactly once.  The grid is said to be *Muirhead-able* if the sum of the entries in each column is the same, but for every  $1 \le i,k \le n-1$ , the sum of the first  $k$  entries in column  $i$  is at least the sum of the first  $k$  entries in column  $i+1$ .  For which  $n$  can one construct a Muirhead-able array such that the entries in each column are decreasing?

*Proposed by Evan Chen*

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