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 each row and column. For any positive integer $k$, a stick is a $1 \times k$ or $k \times 1$ array. For any sieve $A$, let $m(A)$ denote the minimum number of sticks required to partition $A$. Find all possible values of $m(A)$, since $A$ is different in all possible $n \times n$ sieves, for $n=100$.

Let's think step by step and output the final answer within \boxed{}.
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