Ceiling of a uniform ratio being a perfect square

Let $X$ and $Y$ be independent $\text{Unif}(0,1)$ random variables. The chance that the integer $\lceil Y/X\rceil$ equals a perfect square can be written as $q-\pi^2/a$ with $q$ rational and $a$ an integer. Find $aq$.

$33/2$. The value $1$ occurs when $Y\le X$ (probability $\tfrac12$); the value $k^2\ge4$ occurs on a thin triangle of area $\tfrac12\left(\tfrac{1}{k^2-1}-\tfrac{1}{k^2}\right)$. Using the telescoping sum $\sum_{k\ge2}\tfrac{1}{k^2-1}=\tfrac34$ and $\sum_{k\ge2}\tfrac1{k^2}=\tfrac{\pi^2}{6}-1$, the total is $\tfrac{11}{8}-\tfrac{\pi^2}{12}$, so $aq=12\cdot\tfrac{11}{8}$.