Problem: 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).

<image>

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 ∪ \\{t\} does not exceed l. Find an area of F(S, l).

Input

The first line contains two integers n and l (3 ≤ n ≤ 10^4; 1 ≤ l ≤ 8 ⋅ 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 ≤ x_i, y_i ≤ 10^5) — coordinates of polygon's vertices in counterclockwise order. All internal angles of the polygon are strictly less than π. The length l exceeds the perimeter of the polygon by at least 10^{-3}.

Output

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}. 

Examples

Input


3 4
0 0
1 0
0 1


Output


3.012712585980357


Input


4 5
0 0
1 0
1 1
0 1


Output


5.682061989789656


Input


5 17
0 0
2 -1
3 0
4 3
-1 4


Output


37.719371276930820

Note

The following pictures illustrate the example tests.

<image> <image>

<image>

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
template <class T> int sgn(T x) { return (x > 0) - (x < 0); }
template<class T>
struct Point {	typedef Point P;	T x, y;
	explicit Point(T x_=0, T y_=0) : x(x_), y(y_) {}
	bool operator<(P p) const { return tie(x,y) < tie(p.x,p.y); }
	bool operator==(P p) const { return tie(x,y)==tie(p.x,p.y); }
	P operator+(P p) const { return P(x+p.x, y+p.y); }
	P operator-(P p) const { return P(x-p.x, y-p.y); }
	P operator*(T d) const { return P(x*d, y*d); }
	P operator/(T d) const { return P(x/d, y/d); }
	T dot(P p) const { return x*p.x + y*p.y; }
	T cross(P p) const { return x*p.y - y*p.x; }
	T cross(P a, P b) const { return (a-*this).cross(b-*this); }
	T dist2() const { return x*x + y*y; }
	double dist() const { return sqrt((double)dist2()); }
	double angle() const { return atan2(y, x); }
	P unit() const { return *this/dist(); }
	P perp() const { return P(-y, x); }
	P normal() const { return perp().unit(); }
	P rotate(double a) const {	return P(x*cos(a)-y*sin(a),x*sin(a)+y*cos(a)); }
	friend ostream& operator<<(ostream& os, P p) {return os << "(" << p.x << "," << p.y << ")"; }};
template<class P>pair<int, P> lineInter(P s1, P e1, P s2, P e2) {auto d = (e1 - s1).cross(e2 - s2);if (d == 0) return {-(s1.cross(e1, s2) == 0), P(0, 0)};auto p = s2.cross(e1, e2), q = s2.cross(e2, s1);return {1, (s1 * p + e1 * q) / d};}
template<class P>vector<P> circleLine(P c, double r, P a, P b) {P ab = b - a, p = a + ab * (c-a).dot(ab) / ab.dist2();double s = a.cross(b, c), h2 = r*r - s*s / ab.dist2();
	if (h2 < 0) return {};if (h2 == 0) return {p};P h = ab.unit() * sqrt(h2);return {p - h, p + h};}
using P = Point<double>;
int main() {	using namespace std;ios_base::sync_with_stdio(false), cin.tie(nullptr);int N; double L; cin >> N >> L;vector<P> pts(3*N);
	for (int i = 0; i < N; i++) cin >> pts[i].x >> pts[i].y;{
		P center = pts[0];	double ang = (pts[0] - pts[N-1]).angle();	for (int i = 0; i < N; i++) {pts[i] = pts[i] - center;	pts[i] = pts[i].rotate(-ang);}}
	for (int i = 0; i < N; i++) pts[2*N+i] = pts[N+i] = pts[i];for (int i = 0; i < N; i++) L -= (pts[i+1] - pts[i]).dist();assert(L > 0);int lo = 0;int hi = 1;
	P cur_st;	while (true) {	L += (pts[hi] - pts[hi-1]).dist();	double mi = 0;	double ma = L;	for (int z = 0; z < 50; z++) {	double md = (mi + ma) / 2;
			if (md + (pts[hi] - P(md, 0)).dist() <= L) {	mi = md;	} else ma = md;	}cur_st = P(mi, 0);	if (cur_st.cross(pts[hi], pts[hi+1]) >= 0) break;else {	hi++;	continue;	}	}
	auto seg_val = [&](P a, P b) -> double {return (b.x - a.x) * (b.y + a.y) / 2;};double ans = 0;while (lo != N) {	P center = (pts[lo] + pts[hi]) / 2;
		P dir = pts[lo] - pts[hi];	double ang = dir.angle();	double major = L / 2;	double minor = sqrt(major * major - dir.dist2() / 4);
		auto transform_pt = [&](P p) -> P {p = p - center;	p = p.rotate(-ang);	p.x /= major;p.y /= minor;return p;};
		auto untransform_pt = [&](P p) -> P {p.y *= minor;	p.x *= major;	p = p.rotate(ang);	p = p + center;	return p;};
		auto untransform_area = [&](double d) -> double {	return d * minor * major;};	auto get_circ = [&](P a, P b) -> P {vector<P> res = circleLine(P(0,0), 1, a, b);
			assert(!res.empty());if (res.size() >= 2 && res[1].y > res[0].y) swap(res[0], res[1]);	return res[0];};
		P plo = transform_pt(pts[lo]);	P plo1 = transform_pt(pts[lo+1]);	P phi = transform_pt(pts[hi]);	P phi1 = transform_pt(pts[hi+1]);	P st = transform_pt(cur_st);
		P cndlo = get_circ(plo, plo1);	P cndhi = get_circ(phi, phi1);	double alo = (cndlo*-1).perp().angle();	double ahi = (cndhi*-1).perp().angle();
		double ast = (st*-1).perp().angle();bool do_lo = hi+1 == lo+N || (lo+1 < hi && alo < ahi);P nxt_st = untransform_pt(do_lo ? cndlo : cndhi);
		double cur_val = untransform_area(((do_lo ? alo : ahi) - ast) / 2);	cur_val += seg_val(nxt_st, center) + seg_val(center, cur_st);
		if (do_lo) {L -= (pts[lo+1] - pts[lo]).dist();lo++;	} else {L += (pts[hi+1] - pts[hi]).dist();	hi++;}cur_st = nxt_st;ans += cur_val;}
	cout << fixed << setprecision(20) << ans << '\n';return 0;}