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 <iostream>
#include <vector>
#include <string>
#include <stack>
#include <queue>
#include <deque>
#include <map>
#include <set>
#include <tuple>
#include <bitset>
#include <memory>
#include <cmath>
#include <algorithm>
#include <functional>
#include <iomanip>
#include <numeric>
#include <climits>
#include <cfloat>

struct Angle {
	long double sin, cos;
	Angle operator+(const Angle& that) const { return Angle{ sin * that.cos + cos * that.sin, cos * that.cos - sin * that.sin }; }
	Angle operator-(const Angle& that) const { return Angle{ sin * that.cos - cos * that.sin, cos * that.cos + sin * that.sin }; }
	static Angle from_sin(const long double sin);
};
Angle Angle::from_sin(const long double sin) {
	return Angle{ sin, std::sqrt(1 - sin * sin) };
}
struct Vector {
	long double x, y;
	long double cross(const Vector& that) const { return x * that.y - y * that.x; }
	long double dot(const Vector& that) const { return x * that.x + y * that.y; }
	long double length() const { return std::sqrt(x * x + y * y); }
	Angle angle() const { return Angle{ y / length(), x / length() }; }
};
struct Coordinate {
	long double x, y;
	Vector operator-(const Coordinate& that) const { return Vector{ x - that.x, y - that.y }; }
	Coordinate operator+(const Coordinate& that) const { return Coordinate{ x + that.x, y + that.y }; }
	long double distance(const Coordinate& that) const { return (*this - that).length(); }
};
long double inner_area(const long double a, const long double b, const long double c) {
	auto s = (a + b + c) / 2;
	return std::sqrt(s * (s - a) * (s - b) * (s - c));
}
int sign(const long double val) { return (val == 0) ? 0 : (std::signbit(val)) ? -1 : 1; }
struct Edge {
	Coordinate from, to;
	bool is_cross(const Edge& that) const {
		constexpr long double eps = 0.0000000001;
		return sign((to - from).cross(that.to - from)) * sign((to - from).cross(that.from - from)) <= 0 && sign((that.to - that.from).cross(to - that.from)) * sign((that.to - that.from).cross(from - that.from)) <= 0;
	}
	long double length() const {
		return (from - to).length();
	}
	Coordinate operator[](const int index) const {
		return (index == 0) ? from : to;
	}
};
struct Circle {
	Coordinate center;
	long double radius;
	Edge cut(const Circle& that) const {
		auto mid = that.center - center;
		auto catheti = inner_area(radius, that.radius, mid.length()) * 2 / mid.length();
		auto diff_angle = Angle::from_sin(catheti / radius);
		auto mid_angle = mid.angle();
		auto a = mid_angle + diff_angle;
		auto b = mid_angle - diff_angle;
		return Edge{ Coordinate{a.cos * radius, a.sin * radius} + center, Coordinate{b.cos * radius, b.sin * radius} + center };
	}
};
long double min_length(const std::vector<Edge>& edges, const std::pair<int, int> &start, const Coordinate& goal, std::vector<std::vector<long double>>& memo) {
	if (memo[start.first][start.second] >= 0) return memo[start.first][start.second];
	else {
		memo[start.first][start.second] = LDBL_MAX;
		auto current = edges[start.first][start.second];
		auto to_goal = Edge{ current, goal };
		auto is_cross = true;
		for (auto i = start.first + 1; i < edges.size() && is_cross; ++i) {
			if (!edges[i].is_cross(to_goal)) {
				is_cross = false;
			}
		}
		if (is_cross) return memo[start.first][start.second] = to_goal.length();
		for (auto next = start.first + 1; next < edges.size(); ++next) {
			for (auto i = 0; i < 2; ++i) {
				auto to_next = Edge{ current, edges[next][i] };
				is_cross = true;
				for (auto mid = start.first + 1; mid < next; ++mid) {
					if (!edges[mid].is_cross(to_next)) {
						is_cross = false;
					}
				}
				if (is_cross) {
					memo[start.first][start.second] = std::min(memo[start.first][start.second], min_length(edges, std::make_pair(next, i), goal, memo) + to_next.length());
				}
			}
		}
		return memo[start.first][start.second];
	}
}
int main() {
	while (true) {
		int n; std::cin >> n; if (n == 0) return 0;
		std::vector<Circle> circles(n); for (auto& c : circles) std::cin >> c.center.x >> c.center.y >> c.radius;
		std::vector<Edge> edges; edges.push_back(Edge{ circles.front().center, circles.front().center });
		for (auto i = 1; i < n; ++i) {
			edges.push_back(circles[i].cut(circles[i - 1]));
		}
		std::vector<std::vector<long double>> memo(n, std::vector<long double>(2, -1));
		std::cout << std::setprecision(16) << std::fixed << min_length(edges, std::make_pair(0, 0), circles.back().center, memo) << std::endl;
	}
}