Problem: 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.

Examples

Input

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


Output

6
2 3
3 4
4 2


Input

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


Output

6
2 4
3 5
5 1


Input

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


Output

9
2 4
6 3
5 6
4 5

Solution (in CPP):

Solution:
/*{{{*/
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<string>
#include<iostream>
#include<sstream>
#include<set>
#include<map>
#include<queue>
#include<bitset>
#include<vector>
#include<limits.h>
#include<assert.h>
#define SZ(X) ((int)(X).size())
#define ALL(X) (X).begin(), (X).end()
#define REP(I, N) for (int I = 0; I < (N); ++I)
#define REPP(I, A, B) for (int I = (A); I < (B); ++I)
#define FOR(I, A, B) for (int I = (A); I <= (B); ++I)
#define FORS(I, S) for (int I = 0; S[I]; ++I)
#define RS(X) scanf("%s", (X))
#define SORT_UNIQUE(c) (sort(c.begin(),c.end()), c.resize(distance(c.begin(),unique(c.begin(),c.end()))))
#define GET_POS(c,x) (lower_bound(c.begin(),c.end(),x)-c.begin())
#define CASET int ___T; scanf("%d", &___T); for(int cs=1;cs<=___T;cs++)
#define MP make_pair
#define PB push_back
#define MS0(X) memset((X), 0, sizeof((X)))
#define MS1(X) memset((X), -1, sizeof((X)))
#define LEN(X) strlen(X)
#define F first
#define S second
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef long double LD;
typedef pair<int,int> PII;
typedef vector<int> VI;
typedef vector<LL> VL;
typedef vector<PII> VPII;
typedef pair<LL,LL> PLL;
typedef vector<PLL> VPLL;
template<class T> void _R(T &x) { cin >> x; }
void _R(int &x) { scanf("%d", &x); }
void _R(int64_t &x) { scanf("%lld", &x); }
void _R(double &x) { scanf("%lf", &x); }
void _R(char &x) { scanf(" %c", &x); }
void _R(char *x) { scanf("%s", x); }
void R() {}
template<class T, class... U> void R(T &head, U &... tail) { _R(head); R(tail...); }
template<class T> void _W(const T &x) { cout << x; }
void _W(const int &x) { printf("%d", x); }
void _W(const int64_t &x) { printf("%lld", x); }
void _W(const double &x) { printf("%.16f", x); }
void _W(const char &x) { putchar(x); }
void _W(const char *x) { printf("%s", x); }
template<class T,class U> void _W(const pair<T,U> &x) {_W(x.F); putchar(' '); _W(x.S);}
template<class T> void _W(const vector<T> &x) { for (auto i = x.begin(); i != x.end(); _W(*i++)) if (i != x.cbegin()) putchar(' '); }
void W() {}
template<class T, class... U> void W(const T &head, const U &... tail) { _W(head); putchar(sizeof...(tail) ? ' ' : '\n'); W(tail...); }
#ifdef HOME
 #define DEBUG(...) {printf("# ");printf(__VA_ARGS__);puts("");}
#else
 #define DEBUG(...)
#endif
int MOD = 1e9+7;
void ADD(LL& x,LL v){x=(x+v)%MOD;if(x<0)x+=MOD;}
/*}}}*/
const int SIZE = 1e4+10;
int pid;
VPII e[SIZE],pe[SIZE];
int u[SIZE],v[SIZE];
int visit_cnt[SIZE];
int deg[SIZE];
bool visit(int x,int ed,int lt){
    if(x==ed)return 1;
    REP(i,SZ(e[x])){
        int y=e[x][i].F;
        if(y==lt)continue;
        if(visit(y,ed,x)){
            visit_cnt[e[x][i].S]++;
            return 1;
        }
    }
    return 0;
}
int build(int x,int lt,int from_ty){
    int pre=-1;
    REP(i,SZ(e[x])){
        int y=e[x][i].F;
        if(y==lt)continue;
        int nxt=build(y,x,visit_cnt[e[x][i].S]%2);
        if(visit_cnt[e[x][i].S]%2){
            if(pre==-1)pre=nxt;
            else{
                pe[pre].PB(MP(nxt,pid));
                pe[nxt].PB(MP(pre,pid));
                pid++;
                pre=-1;
            }
        }
    }
    if(pre!=-1){
        if(from_ty==0){
            pe[pre].PB(MP(x,pid));
            pe[x].PB(MP(pre,pid));
            pid++;
            return x;
        }
        else return pre;
    }
    return x;
}
int it[SIZE];
bool used[SIZE];
void final_dfs(int x){
    while(it[x]<SZ(pe[x])){
        if(used[pe[x][it[x]].S]){
            it[x]++;
            continue;
        }
        int i=it[x];
        it[x]++;
        used[pe[x][i].S]=1;
        u[pe[x][i].S]=x;
        v[pe[x][i].S]=pe[x][i].F;
        final_dfs(pe[x][i].F);
    }
}
int main(){
    int N,M; R(N,M);
    REPP(i,1,N){
        int a,b;
        R(a,b);
        e[a].PB(MP(b,i));
        e[b].PB(MP(a,i));
    }
    REP(i,M){
        R(u[i],v[i]);
        pe[u[i]].PB(MP(v[i],i));
        pe[v[i]].PB(MP(u[i],i));
        deg[u[i]]++;
        deg[v[i]]++;
        visit(u[i],v[i],u[i]);
    }
    pid=M;
    int an=0;
    REPP(i,1,N){
        an+=min(2,visit_cnt[i]);
    }
    W(an);
    build(1,1,0);
    FOR(i,1,N){
        final_dfs(i);
    }
    REP(i,M)W(u[i],v[i]);
    return 0;
}