Problem: Tony Stark is playing a game with his suits (they have auto-pilot now). He lives in Malibu. Malibu has n junctions numbered from 1 to n, connected with n - 1 roads. One can get from a junction to any other junction using these roads (graph of Malibu forms a tree).

Tony has m suits. There's a special plan for each suit. The i-th suit will appear at the moment of time ti in the junction vi, and will move to junction ui using the shortest path between vi and ui with the speed ci roads per second (passing a junctions takes no time), and vanishing immediately when arriving at ui (if it reaches ui in time q, it's available there at moment q, but not in further moments). Also, suits move continuously (for example if vi ≠ ui, at time <image> it's in the middle of a road. Please note that if vi = ui it means the suit will be at junction number vi only at moment ti and then it vanishes. 

An explosion happens if at any moment of time two suits share the same exact location (it may be in a junction or somewhere on a road; while appearing, vanishing or moving).

Your task is to tell Tony the moment of the the first explosion (if there will be any).

Input

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100 000) — the number of junctions and the number of suits respectively.

The next n - 1 lines contain the roads descriptions. Each line contains two integers ai and bi — endpoints of the i-th road (1 ≤ ai, bi ≤ n, ai ≠ bi).

The next m lines contain the suit descriptions. The i-th of them contains four integers ti, ci, vi and ui (0 ≤ ti ≤ 10 000, 1 ≤ ci ≤ 10 000, 1 ≤ vi, ui ≤ n), meaning the i-th suit will appear at moment of time ti at the junction vi and will move to the junction ui with a speed ci roads per second.

Output

If there would be no explosions at all, print -1 in the first and only line of output.

Otherwise print the moment of the first explosion.

Your answer will be considered correct if its relative or absolute error doesn't exceed 10 - 6.

Examples

Input

6 4
2 5
6 5
3 6
4 6
4 1
27 6 1 3
9 5 1 6
27 4 3 4
11 29 2 6


Output

27.3


Input

6 4
3 1
4 5
6 4
6 1
2 6
16 4 4 5
13 20 6 2
3 16 4 5
28 5 3 5


Output

-1

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
clock_t t = clock();
namespace my_std {
using namespace std;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
template <typename T>
inline T rnd(T l, T r) {
  return uniform_int_distribution<T>(l, r)(rng);
}
template <typename T>
inline bool chkmax(T &x, T y) {
  return x < y ? x = y, 1 : 0;
}
template <typename T>
inline bool chkmin(T &x, T y) {
  return x > y ? x = y, 1 : 0;
}
template <typename T>
inline void read(T &t) {
  t = 0;
  char f = 0, ch = getchar();
  double d = 0.1;
  while (ch > '9' || ch < '0') f |= (ch == '-'), ch = getchar();
  while (ch <= '9' && ch >= '0') t = t * 10 + ch - 48, ch = getchar();
  if (ch == '.') {
    ch = getchar();
    while (ch <= '9' && ch >= '0') t += d * (ch ^ 48), d *= 0.1, ch = getchar();
  }
  t = (f ? -t : t);
}
template <typename T, typename... Args>
inline void read(T &t, Args &...args) {
  read(t);
  read(args...);
}
char __sr[1 << 21], __z[20];
int __C = -1, __zz = 0;
inline void Ot() { fwrite(__sr, 1, __C + 1, stdout), __C = -1; }
inline void print(register int x) {
  if (__C > 1 << 20) Ot();
  if (x < 0) __sr[++__C] = '-', x = -x;
  while (__z[++__zz] = x % 10 + 48, x /= 10)
    ;
  while (__sr[++__C] = __z[__zz], --__zz)
    ;
  __sr[++__C] = '\n';
}
void file() {}
inline void chktime() {}
long long ksm(long long x, int y) {
  long long ret = 1;
  for (; y; y >>= 1, x = x * x)
    if (y & 1) ret = ret * x;
  return ret;
}
}  // namespace my_std
using namespace my_std;
const long double eps = 1e-9;
int n, m;
struct hh {
  int t, nxt;
} edge[101010 << 1];
int head[101010], ecnt;
void make_edge(int f, int t) {
  edge[++ecnt] = (hh){t, head[f]};
  head[f] = ecnt;
  edge[++ecnt] = (hh){f, head[t]};
  head[t] = ecnt;
}
int dfn[101010], top[101010], son[101010], size[101010], fa[101010],
    dep[101010], cnt;
void dfs1(int x, int fa) {
  dep[x] = dep[::fa[x] = fa] + 1;
  size[x] = 1;
  for (int i = head[x]; i; i = edge[i].nxt)
    if (edge[i].t != fa) {
      dfs1(edge[i].t, x);
      size[x] += size[edge[i].t];
      if (size[edge[i].t] > size[son[x]]) son[x] = edge[i].t;
    }
}
void dfs2(int x, int fa, int tp) {
  dfn[x] = ++cnt;
  top[x] = tp;
  if (son[x]) dfs2(son[x], x, tp);
  for (int i = head[x]; i; i = edge[i].nxt)
    if (edge[i].t != fa && edge[i].t != son[x]) dfs2(edge[i].t, x, edge[i].t);
}
inline int dcmp(long double x) { return fabs(x) <= eps ? 0 : (x > 0 ? 1 : -1); }
struct Vector {
  long double x, y;
  Vector(long double xx = 0, long double yy = 0) { x = xx, y = yy; }
  inline const Vector operator+(const Vector &a) const {
    return Vector(x + a.x, y + a.y);
  }
  inline const Vector operator-(const Vector &a) const {
    return Vector(x - a.x, y - a.y);
  }
  inline const Vector operator*(const long double &a) const {
    return Vector(x * a, y * a);
  }
  inline const Vector operator/(const long double &a) const {
    return Vector(x / a, y / a);
  }
  inline const bool operator<(const Vector &a) const {
    return dcmp(x - a.x) == 0 ? dcmp(y - a.y) < 0 : dcmp(x - a.x) < 0;
  }
  inline const bool operator==(const Vector &a) const {
    return dcmp(x - a.x) == 0 && dcmp(y - a.y) == 0;
  }
  inline const bool operator!=(const Vector &a) const {
    return dcmp(x - a.x) != 0 || dcmp(y - a.y) != 0;
  }
};
inline long double Dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
inline long double Cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }
inline long double Len(Vector a) { return sqrt(Dot(a, a)); }
struct Line {
  Vector a, b;
  Line() {}
  Line(Vector aa, Vector bb) { a = aa, b = bb; }
};
vector<Line> vec[101010];
struct hhh {
  long double x;
  int id, type;
  const bool operator<(const hhh &a) const {
    if (dcmp(x - a.x)) return x < a.x;
    return type < a.type;
  }
} ss[101010 * 20];
inline bool OnSeg(Vector p, Line x) {
  Vector a = x.a, b = x.b;
  return dcmp(Cross(p - a, a - b)) == 0 && dcmp(Dot(p - a, p - b)) <= 0;
}
inline Vector LI(Line l1, Line l2) {
  Vector v = l1.a - l2.a, v1 = l1.b - l1.a, v2 = l2.b - l2.a;
  long double t = Cross(v, v2) / Cross(v2, v1);
  return l1.a + v1 * t;
}
inline bool LSI(Line l, Vector a, Vector b) {
  Vector v = l.b - l.a, v1 = a - l.a, v2 = b - l.a;
  return dcmp(Cross(v, v1)) != dcmp(Cross(v, v2));
}
inline long double SSI(Line l1, Line l2) {
  long double ret = 1e9;
  if (OnSeg(l1.a, l2)) chkmin(ret, l1.a.x);
  if (OnSeg(l1.b, l2)) chkmin(ret, l1.b.x);
  if (OnSeg(l2.a, l1)) chkmin(ret, l2.a.x);
  if (OnSeg(l2.b, l1)) chkmin(ret, l2.b.x);
  if (ret < 1e9) return ret;
  if (!LSI(l1, l2.a, l2.b) || !LSI(l2, l1.a, l1.b)) return 1e9;
  return LI(l1, l2).x;
}
int lca(int x, int y) {
  while (top[x] != top[y]) {
    if (dep[top[x]] < dep[top[y]]) swap(x, y);
    x = fa[top[x]];
  }
  return dep[x] > dep[y] ? y : x;
}
void add(int x, int y, long double T, long double V) {
  V = 1.0 / V;
  int l = lca(x, y);
  long double Len = dep[y] - dep[l];
  while (top[x] != top[l]) {
    long double len = dep[x] - dep[top[x]] + 1;
    vec[top[x]].push_back(
        Line(Vector(T, dfn[x]), Vector(T + len * V, dfn[top[x]] - 1)));
    x = fa[top[x]];
    T += len * V;
  }
  long double len = dep[x] - dep[l];
  if (x != l)
    vec[top[x]].push_back(Line(Vector(T, dfn[x]), Vector(T + len * V, dfn[l])));
  T += len * V;
  while (top[y] != top[l]) {
    long double len = dep[y] - dep[top[y]] + 1;
    Len -= len;
    vec[top[y]].push_back(Line(Vector(T + Len * V, dfn[top[y]] - 1),
                               Vector(T + len * V + Len * V, dfn[y])));
    y = fa[top[y]];
  }
  len = dep[y] - dep[l];
  if (y != l)
    vec[top[y]].push_back(Line(Vector(T, dfn[l]), Vector(T + len * V, dfn[y])));
  if (x == l && y == l)
    vec[top[l]].push_back(Line(Vector(T, dfn[l]), Vector(T, dfn[l])));
}
vector<Line> s;
long double T;
struct cmp {
  const bool operator()(const int &x, const int &y) const {
    Line l1 = s[x], l2 = s[y];
    long double y1;
    if (dcmp(l1.b.x - l1.a.x))
      y1 = (l1.a + (l1.b - l1.a) * (T - l1.a.x) / (l1.b.x - l1.a.x)).y;
    else
      y1 = l1.a.y;
    long double y2;
    if (dcmp(l2.b.x - l2.a.x))
      y2 = (l2.a + (l2.b - l2.a) * (T - l2.a.x) / (l2.b.x - l2.a.x)).y;
    else
      y2 = l2.a.y;
    return y1 < y2;
  }
};
set<int, cmp> S;
long double ans = 1e9;
void work() {
  long double ans = 1e9;
  int M = s.size();
  s.push_back(Line(Vector(), Vector()));
  for (int i = (0); i <= (M - 1); i++)
    ss[i * 2] = (hhh){s[i].a.x, i, 0}, ss[i * 2 + 1] = (hhh){s[i].b.x, i, 1};
  sort(ss, ss + M + M);
  for (int i = (0); i <= (M + M - 1); i++) {
    if (ss[i].x > ans) break;
    if (ss[i].type) {
      auto it = S.find(ss[i].id);
      if (it != S.begin() && next(it) != S.end())
        chkmin(ans, SSI(s[*prev(it)], s[*next(it)]));
      S.erase(ss[i].id);
    } else {
      T = ss[i].x;
      s[M] = Line(Vector(T, s[ss[i].id].a.y), Vector(T, s[ss[i].id].a.y));
      if (S.find(M) != S.end()) {
        ans = T;
        break;
      }
      auto it = S.insert(ss[i].id).first;
      if (it != S.begin()) chkmin(ans, SSI(s[*it], s[*prev(it)]));
      if (next(it) != S.end()) chkmin(ans, SSI(s[*it], s[*next(it)]));
    }
  }
  chkmin(::ans, ans);
  S.clear();
}
int main() {
  file();
  read(n, m);
  int x, y;
  for (int i = (1); i <= (n - 1); i++) read(x, y), make_edge(x, y);
  dfs1(1, 0), dfs2(1, 0, 1);
  while (m--) {
    int v, t;
    read(t, v, x, y);
    add(x, y, t, v);
  }
  for (int i = (1); i <= (n); i++)
    if (top[i] == i) s = vec[i], work();
  if (ans < 1e9)
    cout << setprecision(20) << fixed << ans;
  else
    puts("-1");
  return 0;
}