Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1, 2, 3, 4, 5\}$ Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

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