Robot

You are given $n$ points $(x_1, y_1), \dots, (x_n, y_n)$ in the plane. Your goal is to navigate a robot from point $(x_1, y_1)$ to point $(x_n, y_n)$.

From its current point $(x_i, y_i)$, the robot may travel to any other point $(x_j, y_j)$ that is at most $R$ units away, at a speed of $1$ unit per second. Before it moves, however, the robot must turn until it faces $(x_j, y_j)$; this turning happens at a rate of $1$ degree per second.

Compute the shortest time needed for the robot to travel from point $(x_1, y_1)$ to $(x_n, y_n)$. Assume the robot initially faces $(x_n, y_n)$.

To avoid floating-point precision issues, use the double data type rather than float. It is guaranteed that the unrounded shortest time is no more than $0.4$ away from the nearest integer. If you use inverse trigonometric functions, prefer atan2() over acos() or asin().

The input contains multiple test cases. Each test case begins with a line containing two integers $R$ and $n$: $R$ is the maximum distance the robot may travel between points ($10 \le R \le 1000$), and $n$ is the number of points ($2 \le n \le 20$).

Each of the next $n$ lines contains two integers; the $i$-th of these lines contains $x_i$ and $y_i$ ($-1000 \le x_i, y_i \le 1000$). Every point is distinct.

The end of input is marked by a test case with $R = n = -1$.

For each test case, print a single line containing the shortest possible time in seconds (rounded to the nearest integer) required for the robot to travel from $(x_1, y_1)$ to $(x_n, y_n)$. If no such trip is possible, print impossible instead.