Initially, each cell in a $29 \times 29$ table contains the number $0$. A movement consists of selecting a $5 \times 5$ sub-table and adding $+1$ to every cell within this sub-table. Determine the maximum value of $n$ such that, after performing $1000$ movements, there exist four cells whose centers form the vertices of a square, and the sum of the values in these four cells is at least $n$. 

**Note:** The sides of the square do not necessarily have to be parallel to the sides of the table.

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