#include <bits/stdc++.h>
using namespace std;
template <class T>
inline void read(T &x) {
  x = 0;
  char c = getchar();
  int f = 1;
  while (!isdigit(c)) {
    if (c == '-') f = -1;
    c = getchar();
  }
  while (isdigit(c)) {
    x = x * 10 + c - '0';
    c = getchar();
  }
  x *= f;
}
template <class T>
inline void umin(T &x, T y) {
  x = x < y ? x : y;
}
template <class T>
inline void umax(T &x, T y) {
  x = x > y ? x : y;
}
inline unsigned int R() {
  static unsigned int seed = 416;
  return seed ^= seed >> 5, seed ^= seed << 17, seed ^= seed >> 13;
}
const int N = 266666;
const double inf = 1e9, eps = 1e-6;
int n, m, fa[N], top[N], dep[N], sz[N], son[N], dfn[N], idfn[N], num;
vector<int> e[N];
double ans = inf, cur;
struct dot {
  double x, y;
  dot(double X = 0, double Y = 0) {
    x = X;
    y = Y;
  }
};
vector<pair<dot, dot> > mrk[N];
dot a[N], b[N];
int len;
pair<double, int> ev[N];
bool cmp1(pair<double, int> a, pair<double, int> b) {
  return a.first != b.first ? a.first < b.first : a.second > b.second;
}
inline double gety(int i) {
  return a[i].x == b[i].x
             ? a[i].y
             : a[i].y + (cur - a[i].x) / (b[i].x - a[i].x) * (b[i].y - a[i].y);
}
struct node {
  int u;
  bool operator<(const node &a) const {
    return gety(u) != gety(a.u) ? gety(u) < gety(a.u) : u < a.u;
  }
};
set<node> Set;
dot operator-(dot a, dot b) { return dot(a.x - b.x, a.y - b.y); }
double Cross(dot a, dot b) { return a.x * b.y - a.y * b.x; }
bool isin(dot p, dot a, dot b) {
  return p.x >= min(a.x, b.x) - eps && p.x <= max(a.x, b.x) + eps &&
         p.y >= min(a.y, b.y) - eps && p.y <= max(a.y, b.y) + eps;
}
double Abs(double x) { return x > 0 ? x : -x; }
bool on_line(dot p, dot a, dot b) { return Abs(Cross(p - a, b - a)) <= eps; }
double getlen(dot a) { return sqrt(a.x * a.x + a.y * a.y); }
void print(int i) {
  printf("%lf %lf %lf %lf\n", a[i].x, a[i].y, b[i].x, b[i].y);
}
double getans(int i, int j) {
  if (i == j) return inf;
  if (getlen(b[i] - a[i]) > getlen(b[j] - a[j])) swap(i, j);
  if (getlen(b[i] - a[i]) <= eps) {
    if (isin(a[i], a[j], b[j]) && on_line(a[i], a[j], b[j])) return a[i].x;
    return inf;
  }
  dot A = a[i], B = b[i], C = a[j], D = b[j], u = B - A, v = D - C;
  if (Abs(Cross(u, v)) <= eps) {
    if (Abs(gety(i) - gety(j)) <= eps && cur >= A.x - eps && cur <= B.x + eps &&
        cur >= C.x - eps && cur <= D.x + eps) {
      return cur;
    }
    return inf;
  }
  double k = (Cross(D, v) - Cross(A, v)) / Cross(u, v);
  dot p(A.x + k * u.x, A.y + k * u.y);
  if (isin(p, A, B) && isin(p, C, D) && on_line(p, A, B) && on_line(p, C, D))
    return A.x + k * u.x;
  return inf;
}
void upd(int i) {
  set<node>::iterator it = Set.lower_bound((node){i});
  if (it != Set.end()) umin(ans, getans(i, (*it).u));
  if (it != Set.begin()) it--, umin(ans, getans(i, (*it).u));
}
void dfs1(int u, int Fa, int Dep) {
  fa[u] = Fa;
  dep[u] = Dep;
  sz[u] = 1;
  for (register int i = (0); i <= (((int)e[u].size()) - 1); i++)
    if (e[u][i] != Fa) {
      dfs1(e[u][i], u, Dep + 1), sz[u] += sz[e[u][i]];
      if (sz[e[u][i]] > sz[son[u]]) son[u] = e[u][i];
    }
}
void dfs2(int u, int Top) {
  top[u] = Top;
  dfn[u] = ++num;
  idfn[num] = u;
  if (son[u]) dfs2(son[u], Top);
  for (register int i = (0); i <= (((int)e[u].size()) - 1); i++)
    if (e[u][i] != fa[u] && e[u][i] != son[u]) dfs2(e[u][i], e[u][i]);
}
int getlca(int u, int v) {
  while (top[u] != top[v]) {
    if (dep[top[u]] < dep[top[v]]) swap(u, v);
    u = fa[top[u]];
  }
  return dep[u] < dep[v] ? u : v;
}
int getpath(int u, int v, double t, int xs, double c) {
  while (top[u] != top[v]) {
    int x = fa[top[u]];
    mrk[top[u]].push_back(make_pair(
        dot(t, dfn[u]),
        dot(t + xs * (dep[u] - dep[x]) / c, dfn[u] - (dep[u] - dep[x]))));
    t += xs * (dep[u] - dep[x]) / c;
    u = fa[top[u]];
  }
  return u;
}
int main() {
  read(n);
  read(m);
  for (register int i = (1); i <= (n - 1); i++) {
    int u, v;
    read(u);
    read(v);
    e[u].push_back(v);
    e[v].push_back(u);
  }
  dfs1(1, 0, 0);
  dfs2(1, 1);
  for (register int i = (1); i <= (m); i++) {
    int T, C, u, v;
    read(T);
    read(C);
    read(u);
    read(v);
    double t = T, c = C;
    int lca = getlca(u, v);
    int ori = u;
    v = getpath(v, lca, t + (dep[u] + dep[v] - 2 * dep[lca]) * 1.0 / c, -1, c);
    u = getpath(u, lca, t, 1, c);
    t += (dep[ori] - dep[u]) * 1.0 / c;
    if (dep[u] >= dep[v])
      mrk[top[u]].push_back(make_pair(
          dot(t, dfn[u]), dot(t + (dep[u] - dep[v]) * 1.0 / c, dfn[v])));
    else
      mrk[top[u]].push_back(make_pair(
          dot(t, dfn[u]), dot(t + (dep[v] - dep[u]) * 1.0 / c, dfn[v])));
  }
  for (register int t = (1); t <= (n); t++)
    if (((int)mrk[t].size())) {
      Set.clear();
      len = 0;
      for (register int i = (0); i <= (((int)mrk[t].size()) - 1); i++)
        len++, a[len] = mrk[t][i].first, b[len] = mrk[t][i].second;
      for (register int i = (1); i <= (len); i++)
        if (a[i].x > b[i].x) swap(a[i], b[i]);
      for (register int i = (1); i <= (len); i++)
        ev[2 * i - 1] = make_pair(a[i].x, i), ev[2 * i] = make_pair(b[i].x, -i);
      sort(ev + 1, ev + len * 2 + 1, cmp1);
      for (register int i = (1); i <= (len * 2); i++) {
        if ((cur = ev[i].first) >= ans) break;
        if (ev[i].second > 0) {
          set<node>::iterator it = Set.insert((node){ev[i].second}).first, pre,
                              nxt;
          pre = it;
          nxt = it;
          nxt++;
          if (it != Set.begin()) {
            pre--;
            umin(ans, getans((*pre).u, (*it).u));
          }
          if (nxt != Set.end()) {
            umin(ans, getans((*nxt).u, (*it).u));
          }
        } else {
          set<node>::iterator it = Set.find((node){-ev[i].second}), it1, it2;
          it1 = it, it2 = it;
          it2++;
          if (it != Set.begin() && it2 != Set.end()) {
            it1--;
            umin(ans, getans((*it1).u, (*it2).u));
          }
          Set.erase(it);
        }
      }
    }
  if (ans < 1e8)
    printf("%.9lf\n", ans);
  else
    puts("-1");
  return 0;
}

### Prompt

Please create a solution in cpp to the following 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

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
template <class T>
inline void read(T &x) {
  x = 0;
  char c = getchar();
  int f = 1;
  while (!isdigit(c)) {
    if (c == '-') f = -1;
    c = getchar();
  }
  while (isdigit(c)) {
    x = x * 10 + c - '0';
    c = getchar();
  }
  x *= f;
}
template <class T>
inline void umin(T &x, T y) {
  x = x < y ? x : y;
}
template <class T>
inline void umax(T &x, T y) {
  x = x > y ? x : y;
}
inline unsigned int R() {
  static unsigned int seed = 416;
  return seed ^= seed >> 5, seed ^= seed << 17, seed ^= seed >> 13;
}
const int N = 266666;
const double inf = 1e9, eps = 1e-6;
int n, m, fa[N], top[N], dep[N], sz[N], son[N], dfn[N], idfn[N], num;
vector<int> e[N];
double ans = inf, cur;
struct dot {
  double x, y;
  dot(double X = 0, double Y = 0) {
    x = X;
    y = Y;
  }
};
vector<pair<dot, dot> > mrk[N];
dot a[N], b[N];
int len;
pair<double, int> ev[N];
bool cmp1(pair<double, int> a, pair<double, int> b) {
  return a.first != b.first ? a.first < b.first : a.second > b.second;
}
inline double gety(int i) {
  return a[i].x == b[i].x
             ? a[i].y
             : a[i].y + (cur - a[i].x) / (b[i].x - a[i].x) * (b[i].y - a[i].y);
}
struct node {
  int u;
  bool operator<(const node &a) const {
    return gety(u) != gety(a.u) ? gety(u) < gety(a.u) : u < a.u;
  }
};
set<node> Set;
dot operator-(dot a, dot b) { return dot(a.x - b.x, a.y - b.y); }
double Cross(dot a, dot b) { return a.x * b.y - a.y * b.x; }
bool isin(dot p, dot a, dot b) {
  return p.x >= min(a.x, b.x) - eps && p.x <= max(a.x, b.x) + eps &&
         p.y >= min(a.y, b.y) - eps && p.y <= max(a.y, b.y) + eps;
}
double Abs(double x) { return x > 0 ? x : -x; }
bool on_line(dot p, dot a, dot b) { return Abs(Cross(p - a, b - a)) <= eps; }
double getlen(dot a) { return sqrt(a.x * a.x + a.y * a.y); }
void print(int i) {
  printf("%lf %lf %lf %lf\n", a[i].x, a[i].y, b[i].x, b[i].y);
}
double getans(int i, int j) {
  if (i == j) return inf;
  if (getlen(b[i] - a[i]) > getlen(b[j] - a[j])) swap(i, j);
  if (getlen(b[i] - a[i]) <= eps) {
    if (isin(a[i], a[j], b[j]) && on_line(a[i], a[j], b[j])) return a[i].x;
    return inf;
  }
  dot A = a[i], B = b[i], C = a[j], D = b[j], u = B - A, v = D - C;
  if (Abs(Cross(u, v)) <= eps) {
    if (Abs(gety(i) - gety(j)) <= eps && cur >= A.x - eps && cur <= B.x + eps &&
        cur >= C.x - eps && cur <= D.x + eps) {
      return cur;
    }
    return inf;
  }
  double k = (Cross(D, v) - Cross(A, v)) / Cross(u, v);
  dot p(A.x + k * u.x, A.y + k * u.y);
  if (isin(p, A, B) && isin(p, C, D) && on_line(p, A, B) && on_line(p, C, D))
    return A.x + k * u.x;
  return inf;
}
void upd(int i) {
  set<node>::iterator it = Set.lower_bound((node){i});
  if (it != Set.end()) umin(ans, getans(i, (*it).u));
  if (it != Set.begin()) it--, umin(ans, getans(i, (*it).u));
}
void dfs1(int u, int Fa, int Dep) {
  fa[u] = Fa;
  dep[u] = Dep;
  sz[u] = 1;
  for (register int i = (0); i <= (((int)e[u].size()) - 1); i++)
    if (e[u][i] != Fa) {
      dfs1(e[u][i], u, Dep + 1), sz[u] += sz[e[u][i]];
      if (sz[e[u][i]] > sz[son[u]]) son[u] = e[u][i];
    }
}
void dfs2(int u, int Top) {
  top[u] = Top;
  dfn[u] = ++num;
  idfn[num] = u;
  if (son[u]) dfs2(son[u], Top);
  for (register int i = (0); i <= (((int)e[u].size()) - 1); i++)
    if (e[u][i] != fa[u] && e[u][i] != son[u]) dfs2(e[u][i], e[u][i]);
}
int getlca(int u, int v) {
  while (top[u] != top[v]) {
    if (dep[top[u]] < dep[top[v]]) swap(u, v);
    u = fa[top[u]];
  }
  return dep[u] < dep[v] ? u : v;
}
int getpath(int u, int v, double t, int xs, double c) {
  while (top[u] != top[v]) {
    int x = fa[top[u]];
    mrk[top[u]].push_back(make_pair(
        dot(t, dfn[u]),
        dot(t + xs * (dep[u] - dep[x]) / c, dfn[u] - (dep[u] - dep[x]))));
    t += xs * (dep[u] - dep[x]) / c;
    u = fa[top[u]];
  }
  return u;
}
int main() {
  read(n);
  read(m);
  for (register int i = (1); i <= (n - 1); i++) {
    int u, v;
    read(u);
    read(v);
    e[u].push_back(v);
    e[v].push_back(u);
  }
  dfs1(1, 0, 0);
  dfs2(1, 1);
  for (register int i = (1); i <= (m); i++) {
    int T, C, u, v;
    read(T);
    read(C);
    read(u);
    read(v);
    double t = T, c = C;
    int lca = getlca(u, v);
    int ori = u;
    v = getpath(v, lca, t + (dep[u] + dep[v] - 2 * dep[lca]) * 1.0 / c, -1, c);
    u = getpath(u, lca, t, 1, c);
    t += (dep[ori] - dep[u]) * 1.0 / c;
    if (dep[u] >= dep[v])
      mrk[top[u]].push_back(make_pair(
          dot(t, dfn[u]), dot(t + (dep[u] - dep[v]) * 1.0 / c, dfn[v])));
    else
      mrk[top[u]].push_back(make_pair(
          dot(t, dfn[u]), dot(t + (dep[v] - dep[u]) * 1.0 / c, dfn[v])));
  }
  for (register int t = (1); t <= (n); t++)
    if (((int)mrk[t].size())) {
      Set.clear();
      len = 0;
      for (register int i = (0); i <= (((int)mrk[t].size()) - 1); i++)
        len++, a[len] = mrk[t][i].first, b[len] = mrk[t][i].second;
      for (register int i = (1); i <= (len); i++)
        if (a[i].x > b[i].x) swap(a[i], b[i]);
      for (register int i = (1); i <= (len); i++)
        ev[2 * i - 1] = make_pair(a[i].x, i), ev[2 * i] = make_pair(b[i].x, -i);
      sort(ev + 1, ev + len * 2 + 1, cmp1);
      for (register int i = (1); i <= (len * 2); i++) {
        if ((cur = ev[i].first) >= ans) break;
        if (ev[i].second > 0) {
          set<node>::iterator it = Set.insert((node){ev[i].second}).first, pre,
                              nxt;
          pre = it;
          nxt = it;
          nxt++;
          if (it != Set.begin()) {
            pre--;
            umin(ans, getans((*pre).u, (*it).u));
          }
          if (nxt != Set.end()) {
            umin(ans, getans((*nxt).u, (*it).u));
          }
        } else {
          set<node>::iterator it = Set.find((node){-ev[i].second}), it1, it2;
          it1 = it, it2 = it;
          it2++;
          if (it != Set.begin() && it2 != Set.end()) {
            it1--;
            umin(ans, getans((*it1).u, (*it2).u));
          }
          Set.erase(it);
        }
      }
    }
  if (ans < 1e8)
    printf("%.9lf\n", ans);
  else
    puts("-1");
  return 0;
}

```