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.
• 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

• $1 \leq N,M \leq 2000$
• $1 \leq a_i , b_i \leq N$
• $1 \leq u_i , v_i \leq N$
• $a_i \neq b_i$
• $u_i \neq v_i$
• 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.

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

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 $6$.

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

6
2 4
3 5
5 1

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

9
2 4
6 3
5 6
4 5