Contest Duration: - (local time) (130 minutes) Back to Home
E - Walking on a Tree /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

• 2 つの頂点 u_i,v_i が決まっている。
• 高橋君は頂点 u_i から頂点 v_i、または、頂点 v_i から頂点 u_i に向かって、同じ辺は一度しか通らないように移動する。

また、i 回目の散歩で得られる 楽しさ は以下のように定義されています。

• i 回目の散歩で通った辺であって、以下のいずれかの条件を満たすものの個数
• 今回の散歩で初めて通った辺
• 今まで通ったことがあっても、今回とは逆向きでしか通っていなかった辺

### 制約

• 1 ≦ N,M ≦ 2000
• 1 ≦ a_i , b_i ≦ N
• 1 ≦ u_i , v_i ≦ N
• a_i \neq b_i
• u_i \neq v_i
• 入力で与えられるグラフは木である。

### 入力

N M
a_1 b_1
:
a_{N-1} b_{N-1}
u_1 v_1
:
u_M v_M


### 出力

T
u'_1 v'_1
:
u'_M v'_M


ただし、(u'_i,v'_i) は (u_i,v_i) または (v_i,u_i) であり、i 回目の散歩で u'_i から v'_i に向かって移動することを表している。

### 入力例 1

4 3
2 1
3 1
4 1
2 3
3 4
4 2


### 出力例 1

6
2 3
3 4
4 2


### 入力例 2

5 3
1 2
1 3
3 4
3 5
2 4
3 5
1 5


### 出力例 2

6
2 4
3 5
5 1


### 入力例 3

6 4
1 2
2 3
1 4
4 5
4 6
2 4
3 6
5 6
4 5


### 出力例 3

9
2 4
6 3
5 6
4 5


Score : 1500 points

### Problem Statement

Takahashi loves walking on a tree. The tree where Takahashi walks has N vertices numbered 1 through N. The i-th of the N-1 edges connects Vertex a_i and Vertex b_i.

Takahashi has scheduled M walks. The i-th walk is done as follows:

• The walk involves two vertices u_i and v_i that are fixed beforehand.
• Takahashi will walk from u_i to v_i or from v_i to u_i along the shortest path.

The happiness gained from the i-th walk is defined as follows:

• The happiness gained is the number of the edges traversed during the i-th walk that satisfies one of the following conditions:
• In the previous walks, the edge has never been traversed.
• In the previous walks, the edge has only been traversed in the direction opposite to the direction taken in the i-th walk.

Takahashi would like to decide the direction of each walk so that the total happiness gained from the M walks is maximized. Find the maximum total happiness that can be gained, and one specific way to choose the directions of the walks that maximizes the total happiness.

### Constraints

• 1 ≤ N,M ≤ 2000
• 1 ≤ a_i , b_i ≤ N
• 1 ≤ u_i , v_i ≤ N
• a_i \neq b_i
• u_i \neq v_i
• The graph given as input is a tree.

### Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
:
a_{N-1} b_{N-1}
u_1 v_1
:
u_M v_M


### Output

Print the maximum total happiness T that can be gained, and one specific way to choose the directions of the walks that maximizes the total happiness, in the following format:

T
u^'_1 v^'_1
:
u^'_M v^'_M


where (u^'_i,v^'_i) is either (u_i,v_i) or (v_i,u_i), which means that the i-th walk is from vertex u^'_i to v^'_i.

### Sample Input 1

4 3
2 1
3 1
4 1
2 3
3 4
4 2


### Sample Output 1

6
2 3
3 4
4 2


If we decide the directions of the walks as above, he gains the happiness of 2 in each walk, and the total happiness will be 6.

### Sample Input 2

5 3
1 2
1 3
3 4
3 5
2 4
3 5
1 5


### Sample Output 2

6
2 4
3 5
5 1


### Sample Input 3

6 4
1 2
2 3
1 4
4 5
4 6
2 4
3 6
5 6
4 5


### Sample Output 3

9
2 4
6 3
5 6
4 5