Problem: There is a chain consisting of multiple circles on a plane. The first (last) circle of the chain only intersects with the next (previous) circle, and each intermediate circle intersects only with the two neighboring circles.

Your task is to find the shortest path that satisfies the following conditions.

* The path connects the centers of the first circle and the last circle.
* The path is confined in the chain, that is, all the points on the path are located within or on at least one of the circles.

Figure E-1 shows an example of such a chain and the corresponding shortest path.

<image>

Figure E-1: An example chain and the corresponding shortest path


Input

The input consists of multiple datasets. Each dataset represents the shape of a chain in the following format.

> n
>  x1 y1 r1
>  x2 y2 r2
>  ...
>  xn yn rn
>

The first line of a dataset contains an integer n (3 ≤ n ≤ 100) representing the number of the circles. Each of the following n lines contains three integers separated by a single space. (xi, yi) and ri represent the center position and the radius of the i-th circle Ci. You can assume that 0 ≤ xi ≤ 1000, 0 ≤ yi ≤ 1000, and 1 ≤ ri ≤ 25.

You can assume that Ci and Ci+1 (1 ≤ i ≤ n−1) intersect at two separate points. When j ≥ i+2, Ci and Cj are apart and either of them does not contain the other. In addition, you can assume that any circle does not contain the center of any other circle.

The end of the input is indicated by a line containing a zero.

Figure E-1 corresponds to the first dataset of Sample Input below. Figure E-2 shows the shortest paths for the subsequent datasets of Sample Input.

<image>

Figure E-2: Example chains and the corresponding shortest paths


Output

For each dataset, output a single line containing the length of the shortest chain-confined path between the centers of the first circle and the last circle. The value should not have an error greater than 0.001. No extra characters should appear in the output.

Sample Input


10
802 0 10
814 0 4
820 1 4
826 1 4
832 3 5
838 5 5
845 7 3
849 10 3
853 14 4
857 18 3
3
0 0 5
8 0 5
8 8 5
3
0 0 5
7 3 6
16 0 5
9
0 3 5
8 0 8
19 2 8
23 14 6
23 21 6
23 28 6
19 40 8
8 42 8
0 39 5
11
0 0 5
8 0 5
18 8 10
8 16 5
0 16 5
0 24 5
3 32 5
10 32 5
17 28 8
27 25 3
30 18 5
0


Output for the Sample Input


58.953437
11.414214
16.0
61.874812
63.195179






Example

Input

10
802 0 10
814 0 4
820 1 4
826 1 4
832 3 5
838 5 5
845 7 3
849 10 3
853 14 4
857 18 3
3
0 0 5
8 0 5
8 8 5
3
0 0 5
7 3 6
16 0 5
9
0 3 5
8 0 8
19 2 8
23 14 6
23 21 6
23 28 6
19 40 8
8 42 8
0 39 5
11
0 0 5
8 0 5
18 8 10
8 16 5
0 16 5
0 24 5
3 32 5
10 32 5
17 28 8
27 25 3
30 18 5
0


Output

58.953437
11.414214
16.0
61.874812
63.195179

Solution (in CPP):

Solution:
#include <bits/stdc++.h>

using namespace std;

#define REP(i, n) for (int i=0; i<(n); ++i)
#define RREP(i, n) for (int i=(int)(n)-1; i>=0; --i)
#define FOR(i, a, n) for (int i=(a); i<(n); ++i)
#define RFOR(i, a, n) for (int i=(int)(n)-1; i>=(a); --i)

#define SZ(x) ((int)(x).size())
#define all(x) begin(x),end(x)

#define dump(x) cerr<<#x<<" = "<<(x)<<endl
#define debug(x) cerr<<#x<<" = "<<(x)<<" (L"<<__LINE__<<")"<<endl;

template<class T>
ostream &operator<<(ostream &os, const vector <T> &v) {
    os << "[";
    REP(i, SZ(v)) {
        if (i) os << ", ";
        os << v[i];
    }
    return os << "]";
}

template<class T, class U>
ostream &operator<<(ostream &os, const pair <T, U> &p) {
    return os << "(" << p.first << " " << p.second << ")";
}

template<class T>
bool chmax(T &a, const T &b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template<class T>
bool chmin(T &a, const T &b) {
    if (b < a) {
        a = b;
        return true;
    }
    return false;
}

using ll = long long;
using ull = unsigned long long;
using ld = long double;
using P = pair<int, int>;
using vi = vector<int>;
using vll = vector<ll>;
using vvi = vector<vi>;
using vvll = vector<vll>;

const ll MOD = 1e9 + 7;
const int INF = INT_MAX / 2;
const ll LINF = LLONG_MAX / 2;
const ld EPS = 1e-9;

using Real = double;
const Real PI = acos(-1);

using Point = complex<Real>;
namespace std {
    bool operator<(const Point& a, const Point& b) {
        if (a.real() == b.real()) return a.imag() < b.imag();
        return a.real() < b.real();
    }
}
struct Line {
    Point a, b;
    Line() {}
    Line(const Point& a, const Point& b) : a(a), b(b) {}
    friend ostream& operator<<(ostream& os, Line& l) {
        return os << "[" << l.a << "," << l.b << "]";
    }
};
struct Segment : Line {
    Segment() = default;

    Segment(const Point& a, const Point& b) : Line(a, b) {}
};

inline bool eq(Real a, Real b) { return abs(b - a) < EPS; }

Real radian_to_degree(Real r) {
    return r * 180.0 / PI;
}

Real degree_to_radian(Real d) {
    return d * PI / 180.0;
}

Point rotate(const Point &p, Real theta) {
    return p * polar((Real)1.0, theta);
}

Real cross(const Point& a, const Point& b) {
    return a.real() * b.imag() - a.imag() * b.real();
}

Real dot(const Point& a, const Point& b) {
    return a.real() * b.real() + a.imag() * b.imag();
}

Point projection(const Line& l, const Point& p) {
    Real A = dot(l.b - l.a, p - l.a),
            B = dot(l.a - l.b, p - l.b);
    return (A * l.b + B * l.a) / (A + B);
}

bool parallel(const Line& l1, const Line& l2) {
    return eq(cross(l1.a - l1.b, l2.a - l2.b), 0.0);
}

bool orthogonal(const Line& l1, const Line& l2) {
    return eq(dot(l1.a - l1.b, l2.a - l2.b), 0.0);
}

const int COUNTER_CLOCKWISE = 1,
        CLOCKWISE = -1,
        ONLINE_BACK = 2,
        ONLINE_FRONT = -2,
        ON_SEGMENT = 0;
int ccw(const Point& a, Point b, Point c) {
    b = b - a, c = c - a;
    if (cross(b, c) > EPS) return COUNTER_CLOCKWISE;
    if (cross(b, c) < -EPS) return CLOCKWISE;
    if (dot(b, c) < 0) return ONLINE_BACK;
    if (norm(b) < norm(c)) return ONLINE_FRONT;
    return ON_SEGMENT;
}

bool intersected(const Line& l, const Point& p) {
    return abs(ccw(l.a, l.b, p)) != 1;
}

bool intersected(const Segment& s, const Point& p) {
    return ccw(s.a, s.b, p) == 0;
}

bool intersected(const Line& l, const Segment& s) {
    return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}

bool intersected(const Segment& s1, const Segment& s2) {
    return ccw(s1.a, s1.b, s2.a) * ccw(s1.a, s1.b, s2.b) <= 0 and
           ccw(s2.a, s2.b, s1.a) * ccw(s2.a, s2.b, s1.b) <= 0;
}

Real distance(const Line& l, const Point& p) {
    return abs(p - projection(l, p));
}

Real distance(const Segment& s, const Point& p) {
    Point r = projection(s, p);
    if (intersected(s, r)) return abs(r - p);
    return min(abs(s.a - p), abs(s.b - p));
}

Real distance(const Line &l, const Segment &s) {
    if (intersected(l, s)) return 0;
    return min(distance(l, s.a), distance(l, s.b));
}

Real distance(const Segment& s1, const Segment& s2) {
    if (intersected(s1, s2)) return 0.0;
    return min({ distance(s1, s2.a), distance(s1, s2.b),
                 distance(s2, s1.a), distance(s2, s1.b) });
}

Point crosspoint(const Line& l1, const Line& l2) {
    Real A = cross(l2.a - l1.a, l2.b - l1.a),
            B = cross(l2.b - l1.b, l2.a - l1.b);
    return (A * l1.b + B * l1.a) / (A + B);
}

int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(10);

    for (;;) {
        int n; cin >> n;
        if (n == 0) break;
        vector<Point> p(n);
        vector<double> r(n);
        REP(i, n) {
            double x, y; cin >> x >> y >> r[i];
            p[i] = { x, y };
        }

        vector<array<Point,2>> s(n-1);
        REP(i, n-1) {
            double d = abs(p[i+1] - p[i]);
            double a = acos((r[i]*r[i] + d*d - r[i+1]*r[i+1]) / (2 * r[i] * d));
            double t = arg(p[i+1] - p[i]);
            s[i] = { p[i] + polar(r[i], t + a),
                     p[i] + polar(r[i], t - a)};
        }

        vector<double> dist(2*n, 1e9);

        REP(i, n-1) {
            REP(k, 2) {
                bool valid = true;
                REP(j, i) {
                    if (!intersected(
                            Segment{s[j][0],s[j][1]},
                            Segment{p[0], s[i][k]})) {
                        valid = false;
                        break;
                    }
                }
                if (valid) {
                    chmin(dist[2 * i + k], abs(s[i][k] - p[0]));
                }
            }
        }

        REP(i, n-1) {
            REP(ik, 2) {
                FOR(j, i+1, n-1) {
                    REP(jk, 2) {
                        bool valid = true;
                        FOR(l, i+1, j) {
                            if (!intersected(
                                    Segment{ s[l][0], s[l][1] },
                                    Segment{ s[i][ik], s[j][jk] }
                                    )) {
                                valid = false;
                                break;
                            }
                        }
                        if (valid) {
                            chmin(dist[2 * j + jk], dist[2 * i + ik] + abs(s[i][ik] - s[j][jk]));
                        }
                    }
                }
            }
        }

        REP(i, n-1) {
            REP(k, 2) {
                bool valid = true;
                FOR(j, i+1, n-1) {
                    if (!intersected(
                            Segment{s[j][0],s[j][1]},
                            Segment{p[n-1], s[i][k]})) {
                        valid = false;
                        break;
                    }
                }
                if (valid) {
                    chmin(dist[2 * n - 1], dist[2 * i + k] + abs(s[i][k] - p[n-1]));
                }
            }
        }

        bool valid = true;
        REP(i, n-1) {
            if (!intersected(
                    Segment{s[i][0], s[i][1]},
                    Segment{p[0], p[n - 1]}
            )) {
                valid = false;
                break;
            }
        }
        if (valid) {
            chmin(dist[2 * n - 1], abs(p[0] - p[n-1]));
        }

        cout << dist[2 * n - 1] << endl;
    }

    return 0;
}