Planet Hunting

Consider a fictitious solar system made up of a star $S$, a planet $P$, and its moon $M$. Planet $P$ travels around star $S$ on a perfect circular orbit with a revolution period of exactly $T$ Earth days. Moon $M$ travels around planet $P$ on a perfect circular orbit whose revolution period is unknown.

Given the position of moon $M$ relative to star $S$ at three different times, your goal is to compute the distance from star $S$ to planet $P$.

Work in a two-dimensional Cartesian coordinate system whose origin is star $S$. Assume that $P$'s counterclockwise orbit around $S$ and $M$'s counterclockwise orbit around $P$ both lie entirely in the $xy$-plane. Let $(x_1, y_1)$ be the position of moon $M$ at the first observation, $(x_2, y_2)$ its position $k_1$ Earth days later, and $(x_3, y_3)$ its position $k_2$ Earth days after the second observation.

The input contains several test cases. Each test case consists of two lines. The first line contains the integers $T$, $k_1$, and $k_2$ with $1 \le T, k_1, k_2 \le 1000$. The second line contains six floating-point values $x_1$, $y_1$, $x_2$, $y_2$, $x_3$, and $y_3$. The input points are chosen so that the solution is unique, and the final distance from planet $P$ to star $S$ is always within $0.1$ of the nearest integer. The end of input is marked by a single line containing 0 0 0.

For each test case, print the distance from planet $P$ to star $S$, rounded to the nearest integer, on its own line.