Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices. The vertices are numbered 1 to N, and the i-th edge connects Vertex a_i and Vertex b_i.

Takahashi loves the number 3. He is seeking a permutation p_1, p_2, \ldots , p_N of integers from 1 to N satisfying the following condition:

For every pair of vertices (i, j), if the distance between Vertex i and Vertex j is 3, the sum or product of p_i and p_j is a multiple of 3.

Here the distance between Vertex i and Vertex j is the number of edges contained in the shortest path from Vertex i to Vertex j.

Help Takahashi by finding a permutation that satisfies the condition.

Constraints

2\leq N\leq 2\times 10^5

1\leq a_i, b_i \leq N

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

If no permutation satisfies the condition, print -1.

Otherwise, print a permutation satisfying the condition, with space in between.
If there are multiple solutions, you can print any of them.

Sample Input 1

5
1 2
1 3
3 4
3 5

Sample Output 1

1 2 5 4 3

The distance between two vertices is 3 for the two pairs (2, 4) and (2, 5).

p_2 + p_4 = 6

p_2\times p_5 = 6

Thus, this permutation satisfies the condition.

Solution in Python (3.4.3) (judge verdict: Runtime Error)

N = int(input())
T = [[] for _ in range(N)]
for _ in range(N - 1):
    a, b = map(int, input().split())
    a, b = a - 1, b - 1
    T[a].append(b)
    T[b].append(a)


colors = [-1] * N
stack = [(0, 0)]
while stack:
    n, color = stack.pop()
    colors[n] = color
    for to in T[n]:
        if colors[to] != -1:
            continue
        stack.append((to, color ^ 1))


# あまり１、あまり２、あまり０になるように並び替え
X = [[] for _ in range(3)]
for n in range(1, N + 1):
    X[n % 3].append(n)
X.append(X[0])
del X[0]


# 少ない色の方は優先的に３の倍数を使うようにして塗る
ans = []
for color in colors:
    if X[color ^ 1]:
        ans.append(X[color ^ 1].pop())
    else:
        ans.append(X[2].pop())

print(' '.join(map(str, ans)))