Robot Vacuum

A robot vacuum must recognize the layout of a room before it starts cleaning. Its sensor can only recognize a wall that the robot has physically bumped into, so to recognize every wall the robot must bump into all of the room's walls.

The room is shaped like a convex polygon. Given a room with $N$ walls and the robot's starting position $P$, find the length of the shortest path along which the robot bumps into every wall and then returns to its start $P$. If the robot bumps into a vertex of the polygon, it counts as bumping into both walls that meet at that vertex.

The input consists of several test cases.

The first line of each test case contains the number of polygon vertices $N$ ($3 \le N \le 100$) and the coordinates $P_x$, $P_y$ of the robot's starting position ($-10\,000 \le P_x, P_y \le 10\,000$). Each of the next $N$ lines contains the coordinates $x$, $y$ of a wall (polygon) vertex ($-10\,000 \le x, y \le 10\,000$).

The vertices are given in counterclockwise order and every interior angle is less than $180^\circ$. The polygon's edges do not intersect one another, and the robot's starting position is strictly inside the polygon (never on an edge).

For each test case, print one line in the form Case x: L, where $x$ is the test case number starting from $1$ and $L$ is the length of the shortest path, rounded to two decimal places (rounding at the third decimal digit).