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 PYTHON3):

Solution:
import math


class Vec:
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __mul__(self, other):
        return Vec(self.x * other, self.y * other)

    def length(self):
        return math.sqrt(self.x * self.x + self.y * self.y)

    def __str__(self):
        return "({},{})".format(self.x, self.y)

    def __sub__(self, other):
        return Vec(self.x - other.x, self.y - other.y)

    def __add__(self, other):
        return Vec(self.x + other.x, self.y + other.y)

    def __truediv__(self, other):
        return Vec(self.x / other, self.y / other)

    def turn90(self):
        return Vec(-self.y, self.x)

    def normalized(self):
        return self / self.length()

    def is_clock(self, other):
        return self.x * other.y - self.y * other.x < 0


class Circle:
    def __init__(self, x, y, r):
        self.r = r
        self.center = Vec(x, y)

    def intersection(self, other):
        d = (self.center - other.center).length()
        r1 = self.r
        r2 = other.r
        d1 = (d * d + r1 * r1 - r2 * r2) / 2 / d
        x = math.sqrt(r1 * r1 - d1 * d1)
        e1 = (other.center - self.center).normalized()
        e2 = e1.turn90()

        p1 = self.center + e1 * d1 + e2 * x
        p2 = self.center + e1 * d1 - e2 * x
        return p1, p2

    def __str__(self):
        return "({},{})".format(self.center, self.r)


def solve(ps: [Circle]):
    intersects = []
    for i in range(len(ps) - 1):
        ci = ps[i]
        ci_next = ps[i + 1]
        intersection1, intersection2 = ci.intersection(ci_next)
        intersects.append((intersection1, intersection2))

    res = [[ps[0].center, -1, 0]]  # 候補座標とその深度と左右(0,1)
    dists = [[2 ** 32, 2 ** 32] for _ in range(len(intersects))]  # 各交点の距離バッファ

    min_path_length = 2 ** 32

    while len(res) != 0:
        res = sorted(res, key=lambda a: a[1], reverse=True)

        c, depth, lr = res.pop()
        current_path_length = dists[depth][lr] if depth != -1 else 0
        if depth == len(intersects) - 1:
            last_path = ps[-1].center - c
            min_path_length = min(min_path_length, current_path_length + last_path.length())
            continue

        l_limit_min, r_limit_min = [a - c for a in intersects[depth + 1]]  # 左右の限界
        l_limit, r_limit = l_limit_min, r_limit_min  # 到達可能限界
        l_limit_d, r_limit_d = depth + 1, depth + 1

        flag = True
        for i in range(depth + 2, len(intersects)):
            l_limit2, r_limit2 = [a - c for a in intersects[i]]

            if l_limit_min.is_clock(l_limit2):  # 限界更新
                l_limit_min = l_limit2
                if l_limit2.is_clock(r_limit_min):  # 到達可能なら、こっちも更新
                    l_limit = l_limit2
                    l_limit_d = i
            if r_limit2.is_clock(r_limit_min):  # 限界更新
                r_limit_min = r_limit2
                if l_limit_min.is_clock(r_limit2):  # 到達可能なら、こっちも更新
                    r_limit = r_limit2
                    r_limit_d = i
            if r_limit_min.is_clock(l_limit_min):
                flag = False
                break
        last_path = ps[-1].center - c
        if flag and l_limit_min.is_clock(last_path) and last_path.is_clock(r_limit_min):  # この点からゴールまで行ける
            min_path_length = min(min_path_length, current_path_length + last_path.length())
            continue
        if dists[l_limit_d][0] > l_limit.length() + current_path_length:
            dists[l_limit_d][0] = l_limit.length() + current_path_length
            res.append([l_limit + c, l_limit_d, 0])
        if dists[r_limit_d][1] > r_limit.length() + current_path_length:
            dists[r_limit_d][1] = r_limit.length() + current_path_length
            res.append([r_limit + c, r_limit_d, 1])
    return min_path_length


def main():
    while True:
        n = int(input())
        if n == 0:
            break

        ps = [Circle(*[int(a) for a in input().split()]) for _ in range(n)]

        print(solve(ps))


if __name__ == '__main__':
    main()