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

Difficulty: UNKNOWN_DIFFICULTY

Starter code:

Solutions:
def solve():
	from math import acos
	from cmath import phase, rect, pi
	from sys import stdin
	file_input = stdin
	while True:
		n = int(file_input.readline())
		if n == 0:
			break
		C = (map(int, file_input.readline().split()) for i in range(n))
		P = []
		(x, y, r1) = next(C)
		c1 = x + y * 1j
		P.append(c1)
		for (x, y, r2) in C:
			c2 = x + y * 1j
			base = c2 - c1
			d = abs(base)
			a = acos((r1 ** 2 + d ** 2 - r2 ** 2) / (2 * r1 * d))
			t = phase(base)
			cp1 = c1 + rect(r1, t + a)
			cp2 = c1 + rect(r1, t - a)
			P.append(cp1)
			P.append(cp2)
			(c1, r1) = (c2, r2)
		lim = 5000
		dist = [lim] * (2 * n)
		dist[0] = 0
		goal = c1
		g_idx = 2 * n - 1
		indices = ((i + i % 2 + 1, i + i % 2 + 2) for i in range(g_idx))
		for (tpl_idx, cp, d) in zip(indices, P, dist):
			(j, k) = tpl_idx
			s1 = None
			s2 = None
			p_s1 = None
			p_s2 = None
			for (l, cp1, cp2) in zip(range(j, g_idx, 2), P[j::2], P[k::2]):
				t_s1 = cp1 - cp
				t_s2 = cp2 - cp
				if s1 is None or phase(s1 / t_s1) >= 0:
					s1 = t_s1
				if s2 is None or phase(s2 / t_s2) <= 0:
					s2 = t_s2
				if phase(s1 / s2) < 0:
					break
				if p_s1 != s1:
					dist[l] = min(dist[l], d + abs(s1))
					p_s1 = s1
				if p_s2 != s2:
					dist[l + 1] = min(dist[l + 1], d + abs(s2))
					p_s2 = s2
			else:
				gs = goal - cp
				if s1 is None and s2 is None or (phase(s1 / gs) >= 0 and phase(s2 / gs) <= 0):
					dist[g_idx] = min(dist[g_idx], d + abs(gs))
		print(dist[g_idx])
solve()


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]]
	dists = [[2 ** 32, 2 ** 32] for _ in range(len(intersects))]
	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]]
		(l_limit, r_limit) = (l_limit_min, r_limit_min)
		(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):
				l_limit_min = l_limit2
				if l_limit2.is_clock(r_limit_min):
					l_limit = l_limit2
					l_limit_d = i
			if r_limit2.is_clock(r_limit_min):
				r_limit_min = r_limit2
				if l_limit_min.is_clock(r_limit2):
					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):
			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))
main()


import math, string, itertools, fractions, heapq, collections, re, array, bisect, sys, random, time, copy, functools
sys.setrecursionlimit(10 ** 7)
inf = 10 ** 20
eps = 1.0 / 10 ** 13
mod = 10 ** 9 + 7
dd = [(-1, 0), (0, 1), (1, 0), (0, -1)]
ddn = [(-1, 0), (-1, 1), (0, 1), (1, 1), (1, 0), (1, -1), (0, -1), (-1, -1)]

def LI():
	return [int(x) for x in sys.stdin.readline().split()]

def LI_():
	return [int(x) - 1 for x in sys.stdin.readline().split()]

def LF():
	return [float(x) for x in sys.stdin.readline().split()]

def LS():
	return sys.stdin.readline().split()

def I():
	return int(sys.stdin.readline())

def F():
	return float(sys.stdin.readline())

def S():
	return input()

def pf(s):
	return print(s, flush=True)

def _kosa(a1, a2, b1, b2):
	(x1, y1) = a1
	(x2, y2) = a2
	(x3, y3) = b1
	(x4, y4) = b2
	tc = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3)
	td = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4)
	return tc * td < 0

def kosa(a1, a2, b1, b2):
	return _kosa(a1, a2, b1, b2) and _kosa(b1, b2, a1, a2)

def distance(x1, y1, x2, y2):
	return math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)

def distance_p(a, b):
	return distance(a[0], a[1], b[0], b[1])

def polar(a, r):
	return (math.cos(r) * a, math.sin(r) * a)

def intersection_c(ap, bp):
	(ax, ay, ar) = ap
	(bx, by, br) = bp
	d = distance(ax, ay, bx, by)
	if d > ar + br or d < abs(ar - br):
		return None
	ac = math.acos((ar ** 2 + d ** 2 - br ** 2) / (2 * ar * d))
	t = math.atan2(by - ay, bx - ax)
	r1 = polar(ar, t + ac)
	r2 = polar(ar, t - ac)
	return [(ax + r1[0], ay + r1[1]), (ax + r2[0], ay + r2[1])]

def main():
	rr = []

	def f(n):
		a = [LI() for _ in range(n)]
		inters = []
		for i in range(n - 1):
			inters.append(intersection_c(a[i], a[i + 1]))
		dp = [[inf] * 2 for _ in range(n - 1)]
		s = a[0][:2]
		rt = a[-1][:2]
		rf = True
		for k in range(n - 1):
			if not kosa(s, rt, inters[k][0], inters[k][1]):
				rf = False
				break
		if rf:
			return distance_p(s, rt)
		for i in range(n - 1):
			for j in range(2):
				t = inters[i][j]
				f = True
				for k in range(i):
					if not kosa(s, t, inters[k][0], inters[k][1]):
						f = False
						break
				if f:
					dp[i][j] = distance_p(s, t)
		r = inf
		for i in range(n - 1):
			for j in range(2):
				s = inters[i][j]
				sd = dp[i][j]
				for k in range(i + 1, n - 1):
					for l in range(2):
						t = inters[k][l]
						f = True
						for m in range(i + 1, k):
							if not kosa(s, t, inters[m][0], inters[m][1]):
								f = False
								break
						if f:
							nd = sd + distance_p(s, t)
							if dp[k][l] > nd:
								dp[k][l] = nd
				t = a[-1][:2]
				f = True
				for k in range(i + 1, n - 1):
					if not kosa(s, t, inters[k][0], inters[k][1]):
						f = False
						break
				if f:
					nd = sd + distance_p(s, t)
					if r > nd:
						r = nd
		return '{:0.5f}'.format(r)
	while 1:
		n = I()
		if n == 0:
			break
		rr.append(f(n))
	return '\n'.join(map(str, rr))
print(main())


from heapq import heappop, heappush

def cross(a, b):
	return a[0] * b[1] - a[1] * b[0]

def calc(r1, r2):
	a = ((r1[0] - r2[0]) ** 2 + (r1[1] - r2[1]) ** 2) ** 0.5
	b = r1[2]
	c = r2[2]
	cos = (a ** 2 + b ** 2 - c ** 2) / (2 * a * b)
	s = b * cos
	t = (b ** 2 - s ** 2) ** 0.5
	e = ((r2[0] - r1[0]) / a, (r2[1] - r1[1]) / a)
	h = (-e[1], e[0])
	return [[r1[0] + s * e[0] + t * h[0], r1[1] + s * e[1] + t * h[1]], [r1[0] + s * e[0] - t * h[0], r1[1] + s * e[1] - t * h[1]]]

def check(a, b, s, t):
	v = (b[0] - a[0], b[1] - a[1])
	v1 = (s[0] - a[0], s[1] - a[1])
	v2 = (t[0] - a[0], t[1] - a[1])
	if cross(v, v1) * cross(v, v2) > 0:
		return 0
	v = (t[0] - s[0], t[1] - s[1])
	v1 = (a[0] - s[0], a[1] - s[1])
	v2 = (b[0] - s[0], b[1] - s[1])
	if cross(v, v1) * cross(v, v2) > 0:
		return 0
	return 1

def dijkstra(m, v):
	dist = [float('inf')] * m
	q = [(0, 0)]
	dist[0] = 0
	while q:
		(dx, x) = heappop(q)
		for (y, d) in v[x]:
			if dist[y] > dx + d:
				dist[y] = dx + d
				heappush(q, (dx + d, y))
	return dist[m - 1]

def solve(n):
	r = [[int(x) for x in input().split()] for i in range(n)]
	p = [[r[0][0], r[0][1]]]
	for i in range(n - 1):
		p += calc(r[i], r[i + 1])
	p.append([r[-1][0], r[-1][1]])
	m = 2 * n
	v = [[] for i in range(m)]
	gate = []
	for i in range(1, n):
		gate.append([p[2 * i - 1], p[2 * i]])
	for i in range(m):
		x = p[i]
		for j in range(i + 1, m):
			y = p[j]
			for (g1, g2) in gate[(i - 1) // 2 + 1:(j - 1) // 2]:
				if not check(x, y, g1, g2):
					break
			else:
				d = ((x[0] - y[0]) ** 2 + (x[1] - y[1]) ** 2) ** 0.5
				v[i].append((j, d))
				v[j].append((i, d))
	print(dijkstra(m, v))
while 1:
	n = int(input())
	if n == 0:
		break
	solve(n)