We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1 \times 1$ squares with an area of $2010 \times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction, find $a + b$.

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