
Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2=2025$ and 2025-2016=9 is a perfect square. (Of the positive integers between l and 10, only 6 and 7 are not squarish.) For a positive integer $N$, let $S(N)$ be the number of squarish integers between l and $N$, inclusive. Find positive constants $\alpha$ and $\beta$ such that $$\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,$$ or show that no such constants exist. Please provide the sum of $12\alpha v$.

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