CF681E.Runaway to a Shadow

Dima is living in a dormitory, as well as some cockroaches.

At the moment 0 Dima saw a cockroach running on a table and decided to kill it. Dima needs exactly T seconds for aiming, and after that he will precisely strike the cockroach and finish it.

To survive the cockroach has to run into a shadow, cast by round plates standing on the table, in T seconds. Shadow casted by any of the plates has the shape of a circle. Shadow circles may intersect, nest or overlap arbitrarily.

The cockroach uses the following strategy: first he equiprobably picks a direction to run towards and then runs towards it with the constant speed v. If at some moment t ≤ T it reaches any shadow circle, it immediately stops in the shadow and thus will stay alive. Otherwise the cockroach is killed by the Dima's precise strike. Consider that the Dima's precise strike is instant.

Determine the probability of that the cockroach will stay alive.

季马住在宿舍里，同时宿舍里还有一些蟑螂。

在时刻 0，季马看见一只蟑螂正在桌子上奔跑，并决定杀死它。季马需要恰好 $T$ 秒来瞄准，之后他将精准地击中蟑螂并将其消灭。

为了存活，蟑螂必须在 $T$ 秒内跑进由桌上圆形盘子投下的阴影区域中。任意一个盘子投下的阴影均为一个圆形。这些阴影圆可以任意相交、嵌套或重叠。

蟑螂采用如下策略：首先，它以等概率随机选择一个奔跑方向；然后，以恒定速度 $v$ 向该方向奔跑。若在某个时刻 $t \leq T$，它到达任意一个阴影圆的内部（含边界），则它立即在阴影中停下，从而存活下来；否则，蟑螂将被季马的精准一击杀死。假设季马的精准一击是瞬时完成的。

请计算蟑螂存活下来的概率。

In the first line of the input the four integers _x_0, _y_0, v, T (|_x_0|, |_y_0| ≤ 109, 0 ≤ v, T ≤ 109) are given — the cockroach initial position on the table in the Cartesian system at the moment 0, the cockroach's constant speed and the time in seconds Dima needs for aiming respectively.

In the next line the only number n (1 ≤ n ≤ 100 000) is given — the number of shadow circles casted by plates.

In the next n lines shadow circle description is given: the i__th of them consists of three integers x__i, y__i, r__i (|x__i|, |y__i| ≤ 109, 0 ≤ r ≤ 109) — the i__th shadow circle on-table position in the Cartesian system and its radius respectively.

Consider that the table is big enough for the cockroach not to run to the table edges and avoid Dima's precise strike.

输入的第一行包含四个整数 $x_0$、$y_0$、$v$、$T$（满足 $|x_0|, |y_0| \leq 10^9$，$0 \leq v, T \leq 10^9$）——分别表示蟑螂在时刻 $0$ 时于笛卡尔坐标系下的初始位置、蟑螂的恒定速度，以及迪马瞄准所需的时间（单位：秒）。

第二行仅包含一个整数 $n$（$1 \leq n \leq 100\,000$）——表示盘子投下的阴影圆的数量。

接下来的 $n$ 行中，每行描述一个阴影圆：第 $i$ 行包含三个整数 $x_i$、$y_i$、$r_i$（满足 $|x_i|, |y_i| \leq 10^9$，$0 \leq r_i \leq 10^9$）——分别表示第 $i$ 个阴影圆在桌面上的笛卡尔坐标位置及其半径。

假设桌面足够大，蟑螂不会跑到桌边以躲避迪马的精准一击。

Print the only real number p — the probability of that the cockroach will stay alive.

Your answer will be considered correct if its absolute or relative error does not exceed 10 - 4.

输出唯一的实数 $p$——即蟑螂存活的概率。

若你的答案的绝对误差或相对误差不超过 $10^{-4}$，则视为正确。

The picture for the first sample is given below.

Red color stands for points which being chosen as the cockroach's running direction will cause him being killed, green color for those standing for survival directions. Please note that despite containing a circle centered in ( - 2, 2) a part of zone is colored red because the cockroach is not able to reach it in one second.

第一个样例的示意图如下所示。

红色表示蟑螂若朝该方向奔跑将被杀死的点；绿色表示蟑螂朝该方向奔跑可存活的点。请注意：尽管以 $(-2, 2)$ 为圆心的圆形区域部分被涂成红色，但这是因为蟑螂无法在一秒内到达该区域的这部分位置。