Problem:
Score : 400 points

Problem Statement

Given is a tree G with N vertices.
The vertices are numbered 1 through N, and the i-th edge connects Vertex a_i and Vertex b_i.

Consider painting the edges in G with some number of colors.
We want to paint them so that, for each vertex, the colors of the edges incident to that vertex are all different.

Among the colorings satisfying the condition above, construct one that uses the minimum number of colors.

Constraints

2 \le N \le 10^5

1 \le a_i \lt b_i \le N

All values in input are integers.

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

Print N lines.

The first line should contain K, the number of colors used.

The (i+1)-th line (1 \le i \le N-1) should contain c_i, the integer representing the color of the i-th edge, where 1 \le c_i \le K must hold.

If there are multiple colorings with the minimum number of colors that satisfy the condition, printing any of them will be accepted.

Sample Input 1

3
1 2
2 3

Sample Output 1

2
1
2

Sample Input 2

8
1 2
2 3
2 4
2 5
4 7
5 6
6 8

Sample Output 2

4
1
2
3
4
1
1
2

Sample Input 3

6
1 2
1 3
1 4
1 5
1 6

Sample Output 3

5
1
2
3
4
5

Solution in Ruby (2.7.1) (judge verdict: Runtime Error)

n, k = gets.chomp.split.map(&:to_i)
p = gets.chomp.split.map(&:to_i)
c = gets.chomp.split.map(&:to_i)
used = Hash.new(false)

cyc = []
n.times do |i|
  cur = i
  if used[cur]
    next
  end

  q = []  
  while !used[cur]
    used[cur] = true
    q << c[cur]
    cur = p[cur] - 1
  end
  cyc << q
end

res = []
cyc.each do |cs|
  l = cs.size
  dp = []
  ct = Array.new(l, 0)

  l.times do |w|
    l.times do |i|
      ct[i] += cs[(i + w) % l]
    end
    dp << ct.max
  end

  if k <= l || dp[-1] < 0
    res << dp.max
  else
    res << dp[-1] * (k / l) + dp[0...(k % l)].max || 0
  end
end
puts res.max