The Goat

Farmer Byteman drove $n$ stakes into the ground of an infinite pasture. Over the next $k$ days, each morning the farmer takes his goat out to the pasture and ties it, with a cord of length $l$, to a stake chosen uniformly at random among the $n$ stakes. During that day the goat eats all the grass within its reach. To the goat's dismay, the grass does not grow back. It can also happen that the absent-minded farmer ties the goat to the same stake on more than one day.

What is the expected value of the area of the pasture whose grass has been eaten after $k$ days?

The first line contains three integers $n$, $k$, and $l$ ($1 \le n, k, l \le 1\,000$), denoting the number of stakes, the number of days, and the length of the cord, respectively. Each of the next $n$ lines contains the coordinates of one stake as a pair of integers $x_i$, $y_i$ ($-1\,000 \le x_i, y_i \le 1\,000$). No two stakes are placed at the same location.

Print a single real number: the expected area of the pasture whose grass the goat eats over the $k$ days, rounded to exactly six digits after the decimal point.

Consider two stakes over two days. If the goat is tied to the same stake on both days, the eaten area equals $\pi$; if it is tied to two different stakes, the area equals $\dfrac{4\pi}{3} + \dfrac{\sqrt{3}}{2}$. Averaging the four equally likely outcomes gives $\dfrac{7\pi}{6} + \dfrac{\sqrt{3}}{4} \approx 4.098204$.