F. Fiber Shape
Imagine a board with \(n\) pins put into it, the \(i\)-th pin is located at \((x_i, y_i)\). For simplicity, we will restrict the problem to the case where the pins are placed in vertices of a convex polygon.Then, take a non-stretchable string of length \(l\), and put it around all the pins. Place a pencil inside the string and draw a curve around the pins, trying to pull the string in every possible direction. The picture below shows an example of a string tied around the pins and pulled by a pencil (a point \(P\)). Your task is to find an area inside this curve. Formally, for a given convex polygon \(S\) and a length \(l\) let's define a fiber shape \(F(S, l)\) as a set of points \(t\) such that the perimeter of the convex hull of \(S \cup \{t\}\) does not exceed \(l\). Find an area of \(F(S, l)\).
The first line contains two integers \(n\) and \(l\) (\(3 \le n \le 10^4\); \(1 \le l \le 8 \cdot 10^5\)) — the number of vertices of the polygon \(S\) and the length of the string. Next \(n\) lines contain integers \(x_i\) and \(y_i\) (\(-10^5 \le x_i, y_i \le 10^5\)) — coordinates of polygon's vertices in counterclockwise order. All internal angles of the polygon are strictly less than \(\pi\). The length \(l\) exceeds the perimeter of the polygon by at least \(10^{-3}\).
Output a single floating-point number — the area of the fiber shape \(F(S, l)\). Your answer will be considered correct if its absolute or relative error doesn't exceed \(10^{-6}\).
The following pictures illustrate the example tests.