#include<bits/stdc++.h>
using namespace std;
typedef complex<double> P;
double eps=1e-10;

double Sqrt(double x){
  return sqrt(max(0.0,x));
}

struct S{
  P p,q;
  S(P p,P q):p(p),q(q){}
};

struct C{
  P p;
  double r;
  C(P p,double r):p(p),r(r){}
  C(double x,double y,double r):p(P(x,y)),r(r){}
};

vector<P> getCrossPoint(C a,C b){
  vector<P> res;
  P base=b.p-a.p;
  double dist=abs(base);
  if(dist>a.r+b.r+eps)return res;
  if(dist<abs(a.r-b.r)-eps)return res;
  double ca = (a.r*a.r+dist*dist-b.r*b.r) / (2.0*a.r*dist);
  double sa = Sqrt(1.0-ca*ca);
  P fi=a.p+base/dist*a.r*P(ca,sa);
  P se=a.p+base/dist*a.r*P(ca,-sa);
  res.push_back(fi);
  res.push_back(se);
  return res;
}

vector<P> getCrossPoint(C a,S b){
  vector<P> res;
  P base=b.q-b.p;
  P v=a.p-b.p;
  P target=b.p+base*real(v/base);
  double h=abs(a.p-target);
  if(a.r+eps<h)return res;
  
  double w=Sqrt(a.r*a.r-h*h);
  double L=abs(b.p-target)-w;
  double R=abs(b.p-target)+w;
  
  if( -eps<L && L<abs(base)+eps )
    res.push_back(b.p+base/abs(base)*L);

  if( -eps<R && R<abs(base)+eps )
    res.push_back(b.p+base/abs(base)*R);
  
  return res;
}

int n,w;
int px[10],py[10],h[10];
vector<C> vec;

double calc(double r,double t){
  if(r<t)return r;
  return Sqrt( r*r - (r-t)*(r-t) );
}

bool check(P p,double r){
  double a=calc(r,w)-eps;
  double b=100-calc(r,w)+eps;
  if(p.real()<a)return false;
  if(p.real()>b)return false;
  if(p.imag()<a)return false;
  if(p.imag()>b)return false;

  for(int i=0;i<n;i++)
    if( abs(p-vec[i].p) < vec[i].r-eps )return false;

  return true;
}

bool check(vector<P> v,double r){
  for(int i=0;i<(int)v.size();i++)
    if( check(v[i],r) )return true;
  return false;
}

bool check(double r){

  vec.clear();
  for(int i=0;i<n;i++){
    vec.push_back( C( px[i], py[i], calc(r,h[i]) ) );
  }
  
  
  double a=calc(r,w);
  double b=100-calc(r,w);

  P pa=P(a,a),pb=P(b,a),pc=P(b,b),pd=P(a,b);
  if(check(pa,r))return true;
  if(check(pb,r))return true;
  if(check(pc,r))return true;
  if(check(pd,r))return true;

  
  
  for(int i=0;i<n;i++){
    vector<P> tmp1=getCrossPoint(vec[i],S(pa,pb));
    vector<P> tmp2=getCrossPoint(vec[i],S(pb,pc));
    vector<P> tmp3=getCrossPoint(vec[i],S(pc,pd));
    vector<P> tmp4=getCrossPoint(vec[i],S(pd,pa));
    if(check(tmp1,r))return true;
    if(check(tmp2,r))return true;
    if(check(tmp3,r))return true;
    if(check(tmp4,r))return true;
    for(int j=0;j<i;j++){
      vector<P> tmp0=getCrossPoint(vec[i],vec[j]);
      if(check(tmp0,r))return true;
    }
  }
  return false;
}

int main(){
  while(1){
    cin>>n>>w;
    if(n==0&&w==0)break;
    for(int i=0;i<n;i++)cin>>px[i]>>py[i]>>h[i];
    double l=0,r=100000,m;
    for(int i=0;i<100;i++){
      m=(l+r)/2.0;
      if(check(m))l=m;
      else r=m;
    }
    printf("%.8f\n",m);
  }
  return 0;
}
### Prompt

Construct a Cpp code solution to the problem outlined:

An open-top box having a square bottom is placed on the floor. You see a number of needles vertically planted on its bottom.

You want to place a largest possible spheric balloon touching the box bottom, interfering with none of the side walls nor the needles.

Java Specific: Submitted Java programs may not use "java.awt.geom.Area". You may use it for your debugging purposes.

Figure H.1 shows an example of a box with needles and the corresponding largest spheric balloon. It corresponds to the first dataset of the sample input below.

<image>
Figure H.1. The upper shows an example layout and the lower shows the largest spheric balloon that can be placed.



Input

The input is a sequence of datasets. Each dataset is formatted as follows.


n w
x1 y1 h1
:
xn yn hn


The first line of a dataset contains two positive integers, n and w, separated by a space. n represents the number of needles, and w represents the height of the side walls.

The bottom of the box is a 100 × 100 square. The corners of the bottom face are placed at positions (0, 0, 0), (0, 100, 0), (100, 100, 0), and (100, 0, 0).

Each of the n lines following the first line contains three integers, xi, yi, and hi. (xi, yi, 0) and hi represent the base position and the height of the i-th needle. No two needles stand at the same position.

You can assume that 1 ≤ n ≤ 10, 10 ≤ w ≤ 200, 0 < xi < 100, 0 < yi < 100 and 1 ≤ hi ≤ 200. You can ignore the thicknesses of the needles and the walls.

The end of the input is indicated by a line of two zeros. The number of datasets does not exceed 1000.

Output

For each dataset, output a single line containing the maximum radius of a balloon that can touch the bottom of the box without interfering with the side walls or the needles. The output should not contain an error greater than 0.0001.

Example

Input

5 16
70 66 40
38 52 20
40 35 10
70 30 10
20 60 10
1 100
54 75 200
1 10
90 10 1
1 11
54 75 200
3 10
53 60 1
61 38 1
45 48 1
4 10
20 20 10
20 80 10
80 20 10
80 80 10
0 0


Output

26.00000
39.00000
130.00000
49.49777
85.00000
95.00000

### Response

```cpp
#include<bits/stdc++.h>
using namespace std;
typedef complex<double> P;
double eps=1e-10;

double Sqrt(double x){
  return sqrt(max(0.0,x));
}

struct S{
  P p,q;
  S(P p,P q):p(p),q(q){}
};

struct C{
  P p;
  double r;
  C(P p,double r):p(p),r(r){}
  C(double x,double y,double r):p(P(x,y)),r(r){}
};

vector<P> getCrossPoint(C a,C b){
  vector<P> res;
  P base=b.p-a.p;
  double dist=abs(base);
  if(dist>a.r+b.r+eps)return res;
  if(dist<abs(a.r-b.r)-eps)return res;
  double ca = (a.r*a.r+dist*dist-b.r*b.r) / (2.0*a.r*dist);
  double sa = Sqrt(1.0-ca*ca);
  P fi=a.p+base/dist*a.r*P(ca,sa);
  P se=a.p+base/dist*a.r*P(ca,-sa);
  res.push_back(fi);
  res.push_back(se);
  return res;
}

vector<P> getCrossPoint(C a,S b){
  vector<P> res;
  P base=b.q-b.p;
  P v=a.p-b.p;
  P target=b.p+base*real(v/base);
  double h=abs(a.p-target);
  if(a.r+eps<h)return res;
  
  double w=Sqrt(a.r*a.r-h*h);
  double L=abs(b.p-target)-w;
  double R=abs(b.p-target)+w;
  
  if( -eps<L && L<abs(base)+eps )
    res.push_back(b.p+base/abs(base)*L);

  if( -eps<R && R<abs(base)+eps )
    res.push_back(b.p+base/abs(base)*R);
  
  return res;
}

int n,w;
int px[10],py[10],h[10];
vector<C> vec;

double calc(double r,double t){
  if(r<t)return r;
  return Sqrt( r*r - (r-t)*(r-t) );
}

bool check(P p,double r){
  double a=calc(r,w)-eps;
  double b=100-calc(r,w)+eps;
  if(p.real()<a)return false;
  if(p.real()>b)return false;
  if(p.imag()<a)return false;
  if(p.imag()>b)return false;

  for(int i=0;i<n;i++)
    if( abs(p-vec[i].p) < vec[i].r-eps )return false;

  return true;
}

bool check(vector<P> v,double r){
  for(int i=0;i<(int)v.size();i++)
    if( check(v[i],r) )return true;
  return false;
}

bool check(double r){

  vec.clear();
  for(int i=0;i<n;i++){
    vec.push_back( C( px[i], py[i], calc(r,h[i]) ) );
  }
  
  
  double a=calc(r,w);
  double b=100-calc(r,w);

  P pa=P(a,a),pb=P(b,a),pc=P(b,b),pd=P(a,b);
  if(check(pa,r))return true;
  if(check(pb,r))return true;
  if(check(pc,r))return true;
  if(check(pd,r))return true;

  
  
  for(int i=0;i<n;i++){
    vector<P> tmp1=getCrossPoint(vec[i],S(pa,pb));
    vector<P> tmp2=getCrossPoint(vec[i],S(pb,pc));
    vector<P> tmp3=getCrossPoint(vec[i],S(pc,pd));
    vector<P> tmp4=getCrossPoint(vec[i],S(pd,pa));
    if(check(tmp1,r))return true;
    if(check(tmp2,r))return true;
    if(check(tmp3,r))return true;
    if(check(tmp4,r))return true;
    for(int j=0;j<i;j++){
      vector<P> tmp0=getCrossPoint(vec[i],vec[j]);
      if(check(tmp0,r))return true;
    }
  }
  return false;
}

int main(){
  while(1){
    cin>>n>>w;
    if(n==0&&w==0)break;
    for(int i=0;i<n;i++)cin>>px[i]>>py[i]>>h[i];
    double l=0,r=100000,m;
    for(int i=0;i<100;i++){
      m=(l+r)/2.0;
      if(check(m))l=m;
      else r=m;
    }
    printf("%.8f\n",m);
  }
  return 0;
}
```