Vampire

At sunrise the sun is a circle of radius $r$. At time $t=0$ its center is at $(0,-r)$ and it moves upward at speed $(0,1)$ per second. Each building is a rectangle sitting on the horizon. Find the last time when the entire sun lies inside the union of the building silhouettes and the region $y \leq 0$.

The input has multiple datasets. Each begins with $r$ and $n$, followed by $n$ lines with $x_{li}, x_{ri}, h_i$. The input ends with $r=n=0$.

For each dataset, print the last blocked time $t$ to four decimal places with absolute error at most $0.001$.