Space Bowling

A new space bowling alley has opened on the planet Plyuk. The field of the bowling alley is an infinite plane on which pins are placed.

Each pin is a tall cylinder whose base is a circle of radius $r$ meters. All pins are identical. The pins are placed according to the following rules. The pins form $n$ rows: the first row holds one pin, the second holds two, and so on. The last, $n$-th row holds $n$ pins. Introduce a coordinate system on the plane so that the unit of length equals one kilometer. The center of the only pin in the first row is at $(0, 0)$. The centers of the pins in the second row are at $(-1, 1)$ and $(1, 1)$. Thus the centers of the pins in the $i$-th row are at the points with coordinates $(-(i - 1), i - 1)$, $(-(i - 3), i - 1)$, $\ldots$, $(i - 1, i - 1)$.

The game proceeds as follows. A ball of radius $q$ meters is used. The player chooses the initial position of the ball's center $(x_c, y_c)$ and the ball's direction vector $(v_x, v_y)$. The ball is then placed at the initial position and moves without stopping in the direction of the vector $(v_x, v_y)$. The ball is considered to have knocked down a pin if at some moment during the ball's motion the ball and the pin share a point. Knocked-down pins do not change the ball's direction and do not knock down neighboring pins as they fall.

The figure shows an example of a pin arrangement for $r = 500$, $n = 4$ and a ball for $q = 1000$, $x_c = -2$, $y_c = -2$, $v_x = 1$, $v_y = 1$.

Write a program that, given the pin radius $r$, the number of pin rows $n$, the ball radius $q$, its initial position $(x_c, y_c)$, and its direction vector $(v_x, v_y)$, determines the number of pins knocked down by the ball.

The first line of the input file contains two integers $r$ and $n$, separated by exactly one space ($1 \le r \le 700$, $1 \le n \le 200\,000$).

The second line of the input file contains an integer $q$ ($1 \le q \le 10^9$).

The third line of the input file contains two integers $x_c$ and $y_c$, separated by exactly one space ($-10^6 \le x_c \le 10^6$, $-10^6 \le y_c$, $1000 \times y_c < -(r + q)$).

The fourth line of the input file contains two integers $v_x$ and $v_y$, separated by exactly one space ($-10^6 \le v_x \le 10^6$, $0 < v_y \le 10^6$).

The output file must contain a single integer: the number of pins knocked down.

The figure below shows which pins are knocked down (such pins are marked with «х»).