#include <bits/stdc++.h>
std::mt19937 rng(
    (int)std::chrono::steady_clock::now().time_since_epoch().count());
const long double eps = 1e-7;
struct PT {
  long double x, y;
  int freq;
  PT(long double xx = 0, long double yy = 0) : x(xx), y(yy) {}
  PT operator+(const PT &p) const { return PT(x + p.x, y + p.y); }
  PT operator-(const PT &p) const { return PT(x - p.x, y - p.y); }
  PT operator*(long double c) const { return PT(x * c, y * c); }
  PT operator/(long double c) const { return PT(x / c, y / c); }
  long double operator*(const PT &p) const { return x * p.x + y * p.y; }
  long double operator%(const PT &p) const { return x * p.y - y * p.x; }
  long double operator!() const { return sqrtl(x * x + y * y); }
};
struct Segment {
  PT p1, p2;
  long double a, b, c;
  Segment() {}
  Segment(PT st, PT en) {
    p1 = st, p2 = en;
    a = -(st.y - en.y);
    b = st.x - en.x;
    c = a * en.x + b * en.y;
  }
  long double plug(long double x, long double y) { return a * x + b * y - c; }
  long double plug(PT p) { return plug(p.x, p.y); }
  bool inLine(PT p) { return (p - p1) % (p2 - p1) == 0; }
  bool inSegment(PT p) {
    return inLine(p) && (p1 - p2) * (p - p2) >= 0 && (p2 - p1) * (p - p1) >= 0;
  }
  PT lineIntersection(Segment s) {
    long double A = a, B = b, C = c;
    long double D = s.a, E = s.b, F = s.c;
    long double x = (long double)C * E - (long double)B * F;
    long double y = (long double)A * F - (long double)C * D;
    long double tmp = (long double)A * E - (long double)B * D;
    x /= tmp;
    y /= tmp;
    return PT(x, y);
  }
};
int w, h;
bool flag;
std::vector<PT> cutPolygon(std::vector<PT> poly, Segment seg) {
  int n = (int)poly.size();
  std::vector<PT> ans;
  flag = false;
  for (int i = 0; i < n; i++) {
    long double z = seg.plug(poly[i]);
    if (z > -eps) {
      ans.push_back(poly[i]);
    }
    long double z2 = seg.plug(poly[(i + 1) % n]);
    if ((z > eps && z2 < -eps) || (z < -eps && z2 > eps)) {
      ans.push_back(seg.lineIntersection(Segment(poly[i], poly[(i + 1) % n])));
      flag = true;
    }
  }
  return ans;
}
Segment getBisector(PT a, PT b) {
  Segment ans(a, b);
  std::swap(ans.a, ans.b);
  ans.b *= -1;
  ans.c = ans.a * (a.x + b.x) * 0.5 + ans.b * (a.y + b.y) * 0.5;
  return ans;
}
std::vector<std::vector<PT>> getVoronoi(std::vector<PT> pts) {
  int n = (int)pts.size();
  std::vector<int> p(n, 0);
  for (int i = 0; i < n; i++) {
    p[i] = i;
  }
  shuffle(p.begin(), p.end(), rng);
  std::vector<std::vector<PT>> ans(n);
  ans[0].emplace_back(0, 0);
  ans[0].emplace_back(w, 0);
  ans[0].emplace_back(w, h);
  ans[0].emplace_back(0, h);
  for (int i = 1; i < n; i++) {
    ans[i] = ans[0];
  }
  for (auto i : p) {
    for (auto j : p) {
      if (j == i) continue;
      ans[j] = cutPolygon(ans[j], getBisector(pts[j], pts[i]));
      if (flag) {
      }
    }
  }
  return ans;
}
int main() {
  std::ios_base::sync_with_stdio(false);
  std::cin.tie(NULL);
  int n;
  std::cin >> w >> h >> n;
  std::vector<PT> pts;
  for (int i = 0; i < n; i++) {
    int x, y;
    std::cin >> x >> y;
    int id = -1;
    for (int j = 0; j < (int)pts.size(); j++) {
      if (int(pts[j].x + 0.5) == x && int(pts[j].y + 0.5) == y) {
        pts[j].freq++;
        id = j;
        break;
      }
    }
    if (id == -1) {
      PT cur(x, y);
      cur.freq = 1;
      pts.push_back(cur);
    }
  }
  long double ans = 0;
  n = (int)pts.size();
  auto voronoi = getVoronoi(pts);
  for (int i = 0; i < n; i++) {
    auto cell = voronoi[i];
    if (pts[i].freq > 1) {
      for (auto end : cell) {
        ans = std::max(ans, !(end - pts[i]));
      }
      continue;
    }
    std::vector<PT> close;
    std::vector<bool> got(n, false);
    for (int k = 0; k < (int)cell.size(); k++) {
      PT curPT = (cell[k] + cell[(k + 1) % (int)cell.size()]) / 2;
      long double cur = 1e9;
      int id = -1;
      for (int j = 0; j < n; j++) {
        if (j == i) continue;
        long double curDist = !(pts[j] - curPT);
        if (curDist < cur) {
          cur = curDist;
          id = j;
        }
      }
      if (got[id]) continue;
      got[id] = true;
      close.push_back(pts[id]);
    }
    for (int k = 0; k < (int)close.size(); k++) {
      auto curPoly = cell;
      for (int kk = 0; kk < (int)close.size(); kk++) {
        if (kk == k) continue;
        curPoly = cutPolygon(curPoly, getBisector(close[k], close[kk]));
      }
      for (auto end : curPoly) {
        ans = std::max(ans, !(close[k] - end));
      }
    }
  }
  std::cout << std::fixed << std::setprecision(15) << ans << std::endl;
}

### Prompt

Your task is to create a cpp solution to the following problem:

Gena doesn't like geometry, so he asks you to solve this problem for him.

A rectangle with sides parallel to coordinate axes contains n dots. Let's consider some point of the plane. Let's count the distances from this point to the given n points. Let's sort these numbers in the non-decreasing order. We'll call the beauty of the point the second element of this array. If there are two mimimum elements in this array, the beaty will be equal to this minimum.

Find the maximum beauty of a point inside the given rectangle.

Input

The first line contains three integers w, h, n (1 ≤ w, h ≤ 106, 2 ≤ n ≤ 1000) — the lengths of the rectangle sides and the number of points. Next n lines contain two integers xi, yi (0 ≤ xi ≤ w, 0 ≤ yi ≤ h) each — the coordinates of a point. It is possible that it will be coincident points.

Output

Print a single number — the maximum beauty of a point with the absolute or relative error of at most 10 - 9.

Examples

Input

5 5 4
0 0
5 0
0 5
5 5


Output

4.99999999941792340


Input

5 5 3
4 0
2 5
4 1


Output

5.65685424744772010

Note

The point which beauty we need to find must have coordinates (x, y), where 0 ≤ x ≤ w, 0 ≤ y ≤ h. Some of the n points can coincide.

### Response

```cpp
#include <bits/stdc++.h>
std::mt19937 rng(
    (int)std::chrono::steady_clock::now().time_since_epoch().count());
const long double eps = 1e-7;
struct PT {
  long double x, y;
  int freq;
  PT(long double xx = 0, long double yy = 0) : x(xx), y(yy) {}
  PT operator+(const PT &p) const { return PT(x + p.x, y + p.y); }
  PT operator-(const PT &p) const { return PT(x - p.x, y - p.y); }
  PT operator*(long double c) const { return PT(x * c, y * c); }
  PT operator/(long double c) const { return PT(x / c, y / c); }
  long double operator*(const PT &p) const { return x * p.x + y * p.y; }
  long double operator%(const PT &p) const { return x * p.y - y * p.x; }
  long double operator!() const { return sqrtl(x * x + y * y); }
};
struct Segment {
  PT p1, p2;
  long double a, b, c;
  Segment() {}
  Segment(PT st, PT en) {
    p1 = st, p2 = en;
    a = -(st.y - en.y);
    b = st.x - en.x;
    c = a * en.x + b * en.y;
  }
  long double plug(long double x, long double y) { return a * x + b * y - c; }
  long double plug(PT p) { return plug(p.x, p.y); }
  bool inLine(PT p) { return (p - p1) % (p2 - p1) == 0; }
  bool inSegment(PT p) {
    return inLine(p) && (p1 - p2) * (p - p2) >= 0 && (p2 - p1) * (p - p1) >= 0;
  }
  PT lineIntersection(Segment s) {
    long double A = a, B = b, C = c;
    long double D = s.a, E = s.b, F = s.c;
    long double x = (long double)C * E - (long double)B * F;
    long double y = (long double)A * F - (long double)C * D;
    long double tmp = (long double)A * E - (long double)B * D;
    x /= tmp;
    y /= tmp;
    return PT(x, y);
  }
};
int w, h;
bool flag;
std::vector<PT> cutPolygon(std::vector<PT> poly, Segment seg) {
  int n = (int)poly.size();
  std::vector<PT> ans;
  flag = false;
  for (int i = 0; i < n; i++) {
    long double z = seg.plug(poly[i]);
    if (z > -eps) {
      ans.push_back(poly[i]);
    }
    long double z2 = seg.plug(poly[(i + 1) % n]);
    if ((z > eps && z2 < -eps) || (z < -eps && z2 > eps)) {
      ans.push_back(seg.lineIntersection(Segment(poly[i], poly[(i + 1) % n])));
      flag = true;
    }
  }
  return ans;
}
Segment getBisector(PT a, PT b) {
  Segment ans(a, b);
  std::swap(ans.a, ans.b);
  ans.b *= -1;
  ans.c = ans.a * (a.x + b.x) * 0.5 + ans.b * (a.y + b.y) * 0.5;
  return ans;
}
std::vector<std::vector<PT>> getVoronoi(std::vector<PT> pts) {
  int n = (int)pts.size();
  std::vector<int> p(n, 0);
  for (int i = 0; i < n; i++) {
    p[i] = i;
  }
  shuffle(p.begin(), p.end(), rng);
  std::vector<std::vector<PT>> ans(n);
  ans[0].emplace_back(0, 0);
  ans[0].emplace_back(w, 0);
  ans[0].emplace_back(w, h);
  ans[0].emplace_back(0, h);
  for (int i = 1; i < n; i++) {
    ans[i] = ans[0];
  }
  for (auto i : p) {
    for (auto j : p) {
      if (j == i) continue;
      ans[j] = cutPolygon(ans[j], getBisector(pts[j], pts[i]));
      if (flag) {
      }
    }
  }
  return ans;
}
int main() {
  std::ios_base::sync_with_stdio(false);
  std::cin.tie(NULL);
  int n;
  std::cin >> w >> h >> n;
  std::vector<PT> pts;
  for (int i = 0; i < n; i++) {
    int x, y;
    std::cin >> x >> y;
    int id = -1;
    for (int j = 0; j < (int)pts.size(); j++) {
      if (int(pts[j].x + 0.5) == x && int(pts[j].y + 0.5) == y) {
        pts[j].freq++;
        id = j;
        break;
      }
    }
    if (id == -1) {
      PT cur(x, y);
      cur.freq = 1;
      pts.push_back(cur);
    }
  }
  long double ans = 0;
  n = (int)pts.size();
  auto voronoi = getVoronoi(pts);
  for (int i = 0; i < n; i++) {
    auto cell = voronoi[i];
    if (pts[i].freq > 1) {
      for (auto end : cell) {
        ans = std::max(ans, !(end - pts[i]));
      }
      continue;
    }
    std::vector<PT> close;
    std::vector<bool> got(n, false);
    for (int k = 0; k < (int)cell.size(); k++) {
      PT curPT = (cell[k] + cell[(k + 1) % (int)cell.size()]) / 2;
      long double cur = 1e9;
      int id = -1;
      for (int j = 0; j < n; j++) {
        if (j == i) continue;
        long double curDist = !(pts[j] - curPT);
        if (curDist < cur) {
          cur = curDist;
          id = j;
        }
      }
      if (got[id]) continue;
      got[id] = true;
      close.push_back(pts[id]);
    }
    for (int k = 0; k < (int)close.size(); k++) {
      auto curPoly = cell;
      for (int kk = 0; kk < (int)close.size(); kk++) {
        if (kk == k) continue;
        curPoly = cutPolygon(curPoly, getBisector(close[k], close[kk]));
      }
      for (auto end : curPoly) {
        ans = std::max(ans, !(close[k] - end));
      }
    }
  }
  std::cout << std::fixed << std::setprecision(15) << ans << std::endl;
}

```