19462번 - Immigration 스페셜 저지다국어

Peter the Pig is riding a tractor, starting at the origin at moment $0$ and moving along the $x$-axis with speed $u$. Since the road is infinite and not interesting enough, Peter is looking for entertainment. At moment $t_0$ he notices some object at point $(x_0, y_0)$ moving with velocity $v_0 = (v_{0x}, v_{0y})$. So, Peter starts to rotate his head always keeping his eyes fixed at the object. From time to time ($n$ times in total), the object changes its velocity. Namely, at moment $t_i$ the object instantly changes its velocity to $v_i = (v_{ix}, v_{iy})$ for $i = 1, 2, \ldots, n$. It is guaranteed that the object never collides with Peter.

Define $f(t)$ as the directed angle (measured in radians) between $x$-axis and the direction from Peter to the object at moment $t$. We allow $f(t)$ to differ from the actual angle by a multiple of $2 \pi$ as long as the function $f(t)$ is continuous. Define the angular speed of Peter's head as the derivative $f'(t)$. You need to find the maximum of $|f'(t)|$ from the moment $t_0$ to the infinity. In other words, you need to find the maximum of the absolute value of angular speed. It is guaranteed that the answer is finite.

The first line contains four integers $u$, $x_0$, $y_0$, $n$ ($1 \le u \le 100$, $|x_0|, |y_0| \le 10^8$, $0 \le n \le 10^5$) --- the speed of Peter's tractor, the coordinates of the object when first observed by Peter, and the number of times the object changes its velocity.

Each of the next $(n + 1)$ lines contains three space-separated integers $t_i$, $v_{ix}$, $v_{iy}$ ($0 \le t_i \le 10^6$, $|v_{ix}|, |v_{iy}| \le 100$). It is guaranteed that $t_0 < t_1 < \ldots < t_n$. Note that it is possible that $(v_{ix}, v_{iy}) = (0, 0)$. Also note that movement of the object after moment $t_n$ is infinite.

In the only line print one real number with absolute or relative error not exceeding $10^{-6}$ --- the maximum absolute value of the angular speed of Peter's head during his observation of the object.