constexpr auto EPS = numeric_limits<double>::epsilon();

struct point {
	ll x, y;
	point() { x = y = 0; }
	point(ll _x, ll _y) : x(_x), y(_y) {}
	bool operator < (point p) const { return (x == p.x && y < p.y) || x < p.x; }
	bool operator == (point p) const { return x == p.x && y == p.y; }
};

double dist(point& p1, point& p2) { return hypot(p1.x - p2.x, p1.y - p2.y); }

struct vec {
	ll x, y;
	vec(ll _x, ll _y) : x(_x), y(_y) {}
	vec(point a, point b) { x = b.x - a.x, y = b.y - a.y; }
};

ll dot(vec a, vec b) { return a.x * b.x + a.y * b.y; }

ll norm_sq(vec v) { return v.x * v.x + v.y * v.y; }

ll cross(vec a, vec b) { return a.x * b.y - a.y * b.x; }

double angle(point a, point o, point b) {
	vec oa(o, a), ob(o, b);
	return acos(dot(oa, ob) / sqrt(norm_sq(oa) * norm_sq(ob)));
}

// (q.x - p.x) * (r.y - p.y) - (r.x - p.x) * (q.y - p.y)
bool ccw(point p, point q, point r) { return cross(vec(p, q), vec(p, r)) > 0; }

struct line {
	ll a, b, c;
	bool operator < (const line& l) const { return tie(a, b, c) < tie(l.a, l.b, l.c); }
	line(point p1, point p2) {
	    a = p2.y - p1.y, b = p1.x - p2.x, c = p1.y * p2.x - p1.x * p2.y;
	    ll d = __gcd(__gcd(abs(a), abs(b)), abs(c));
	    a /= d, b /= d, c /= d;
	}
};

double point_to_line(point p, line l) { return abs(l.a * p.x + l.b * p.y + l.c) / hypot(l.a, l.b); }

line perp_bi(point p1, point p2) {
	ll a = 2 * (p2.x - p1.x), b = 2 * (p2.y - p1.y), c = -a * (p1.x + p2.x) / 2 - b * (p1.y + p2.y) / 2;
	ll d = __gcd(__gcd(abs(a), abs(b)), abs(c));
	return { a / d, b / d, c / d };
}

point reflect(point p, line l) {
	ll x = p.x * (l.b * l.b - l.a * l.a) - 2 * l.a * (l.b * p.y + l.c);
	ll y = p.y * (l.a * l.a - l.b * l.b) - 2 * l.b * (l.a * p.x + l.c);
	ll d = l.a * l.a + l.b * l.b;
	if (x % d && y % d) return { 10001, 10001 };
	else return { x / d, y / d };
}

struct seg {
	point p1, p2;
	seg(point _p1, point _p2) : p1(_p1), p2(_p2) {}
};

line::line(seg s) { line(s.p1, s.p2); }

double point_to_seg(point p, seg s) {
	ll a = norm_sq(to_vec(s.p1, s.p2)), b = norm_sq(to_vec(p, s.p1)), c = norm_sq(to_vec(p, s.p2));
	if (a + b > c && a + c > b) return point_to_line(p, to_line(s));
	else return min(dist(p, s.p1), dist(p, s.p2));
}

bool seg_isect(seg a, seg b) {
	return ccw(a.p1, a.p2, b.p1) * ccw(a.p1, a.p2, b.p2) <= 0 && ccw(b.p1, b.p2, a.p1) * ccw(b.p1, b.p2, a.p2) <= 0);
}