Derek fills a square $10 \times 10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N$. Find the minimum possible value of $N$.

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