
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
Fix an integer  $n \geq 2$ . An  $n\times n$  sieve is an  $n\times n$  array with  $n$  cells removed so that exactly one cell is removed from every row and every column. A stick is a  $1\times k$  or  $k\times 1$  array for any positive integer  $k$ . For any sieve  $A$ , let  $m(A)$  be the minimal number of sticks required to partition  $A$ . Find all possible values of  $m(A)$ , as  $A$  varies over all possible  $n\times n$  sieves. 

*Palmer Mebane*

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