Horse tied outside the castle

The castle wall is a convex polygon with $N$ vertices. A tree stands outside the wall at the integer coordinates $(X, Y)$, and a knight ties his horse to that tree with a rope of length $L$.

The horse cannot enter the castle, and the rope cannot pass through the wall. When the rope catches on a corner of the wall it bends at that vertex, so the horse walks around the castle as long as the rope reaches.

The horse reaches a point $P$ outside the castle exactly when the shortest path from the tree to $P$ that avoids the interior of the castle has length at most $L$.

Compute the area of the region the horse reaches.

The first line contains the coordinates of the tree, $X$ and $Y$, and the length of the rope $L$, separated by spaces. The second line contains the number of vertices $N$. Each of the next $N$ lines contains two integers $X_i$ and $Y_i$, the vertices of the polygon in clockwise order.

Print the area of the region the horse reaches, rounded to exactly five digits after the decimal point. For an area of $\pi$, print 3.14159.

• $3 \le N \le 300$
• $-10000 \le X_i, Y_i \le 10000$
• $-10000 \le X, Y \le 10000$
• $1 \le L \le 100000$
• All coordinates are integers.
• The polygon is convex, and no three vertices lie on one line.
• The tree is outside the castle. It is never inside the wall or on it.
• Let $S$ be the perimeter of the convex hull of the $N$ vertices together with the tree. Then $2L < S$, so the region the horse reaches by going around the castle to the left never overlaps the region it reaches by going around to the right.