The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\]
Find the number of points $P$ with integer coordinates such that the taxicab distance between $P$ and the origin is less than or equal to $20$.

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