Problem: This is the hard version of the problem. The difference between the versions is in the number of possible operations that can be made. You can make hacks if and only if you solved both versions of the problem.

You are given a binary table of size n × m. This table consists of symbols 0 and 1.

You can make such operation: select 3 different cells that belong to one 2 × 2 square and change the symbols in these cells (change 0 to 1 and 1 to 0).

Your task is to make all symbols in the table equal to 0. You are allowed to make at most nm operations. You don't need to minimize the number of operations.

It can be proved, that it is always possible.

Input

The first line contains a single integer t (1 ≤ t ≤ 5000) — the number of test cases. The next lines contain descriptions of test cases.

The first line of the description of each test case contains two integers n, m (2 ≤ n, m ≤ 100).

Each of the next n lines contains a binary string of length m, describing the symbols of the next row of the table.

It is guaranteed, that the sum of nm for all test cases does not exceed 20000.

Output

For each test case print the integer k (0 ≤ k ≤ nm) — the number of operations.

In the each of the next k lines print 6 integers x_1, y_1, x_2, y_2, x_3, y_3 (1 ≤ x_1, x_2, x_3 ≤ n, 1 ≤ y_1, y_2, y_3 ≤ m) describing the next operation. This operation will be made with three cells (x_1, y_1), (x_2, y_2), (x_3, y_3). These three cells should be different. These three cells should belong to some 2 × 2 square.

Example

Input


5
2 2
10
11
3 3
011
101
110
4 4
1111
0110
0110
1111
5 5
01011
11001
00010
11011
10000
2 3
011
101


Output


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

Note

In the first test case, it is possible to make only one operation with cells (1, 1), (2, 1), (2, 2). After that, all symbols will be equal to 0.

In the second test case:

  * operation with cells (2, 1), (3, 1), (3, 2). After it the table will be: 
    
          
    011  
    001  
    000  
    

  * operation with cells (1, 2), (1, 3), (2, 3). After it the table will be: 
    
          
    000  
    000  
    000  
    




In the fifth test case:

  * operation with cells (1, 3), (2, 2), (2, 3). After it the table will be: 
    
          
    010  
    110  
    

  * operation with cells (1, 2), (2, 1), (2, 2). After it the table will be: 
    
          
    000  
    000  
    

Solution (in JAVA):

Solution:
//package cf;
import java.io.*;
import java.util.*;

public class T_class {
    static int m=2147483647;///this is 2^32 -1 (is prime number) and should be used for hashing
    static int p=1000000007;
    static int max=(int)2e3+10;
    static boolean prime[]=new boolean[max+5];
    static  int dp[][]=new int[(int)max][max];
    static  List<Integer> total_prime=new ArrayList<>();
    public static class pr
    {
        int ai;
        int ti;
        pr(int a,int b)
        {
            ai=a;
            ti=b;
        }
    }
    public static void main(String[] args) throws Exception {
        BufferedWriter out = new BufferedWriter(new OutputStreamWriter(new FileOutputStream(FileDescriptor.out), "ASCII"), 512);
        FastReader sc=new FastReader();
        int t=sc.nextInt();
        StringBuilder sb2=new StringBuilder();

        while(t-- >0) {
            int n=sc.nextInt();
            int m=sc.nextInt();
            int mt[][]=new int[n][m];
            for(int i=0;i<n;i++)
            {
                char inp[]=sc.nextLine().toCharArray();
                for(int j=0;j<m;j++)
                {
                    mt[i][j]=inp[j]-'0';
                }
            }
            int tot=0;
            int step[]={0,3,2,1,4};
            int dx[]={0,0,1,1};
            int dy[]={0,1,0,1};
            StringBuilder ins=new StringBuilder();

            for(int i=0;i<n-2;i++)
            {
                for(int j=0;j<=m-1;j++)
                {
                    if(j==m-1)
                    {
                        if(mt[i][j]==0)
                            continue;
                        mt[i][j]=0;
                        mt[i+1][j]=1-mt[i+1][j];
                        mt[i+1][j-1]=1-mt[i+1][j-1];
                        StringBuilder indside=new StringBuilder();
                        indside.append((i+1)+" "+(1+j)+" "+(i+2)+" "+(j+1)+" "+(i+2)+" "+(j));
                        indside.append("\n");
                        ins.append(indside.toString());
                        tot++;
                        continue;
                    }
                    int ar[][]=new int[4][3];
                    int ones=0;
                    int st=1;
                    int end=3;
                    ar[0][0]=i;
                    ar[0][1]=j;
                    ar[0][2]=mt[i][j];
                    for(int k=1;k<=3;k++)
                    {
                        int x=i+dx[k];
                        int y=j+dy[k];
                        if(mt[x][y]==0) {
                            ar[st][0] = x;
                            ar[st][1] = y;
                            ar[st][2] = mt[x][y];
                        st++;
                        }
                        else
                        {
                            ar[end][0] = x;
                            ar[end][1] = y;
                            ar[end][2] = mt[x][y];
                            end--;
                        }
                        ones+=mt[x][y]==1?1:0;
                       // mt[x][y]=0;
                    }
                    if(ar[0][2]==1) {
                        tot++;
                        StringBuilder indside=new StringBuilder();
                        indside.append((ar[0][0]+1)+" "+(1+ar[0][1])+" ");
                        mt[ar[0][0]][ar[0][1]]=0;
                        for(int k=3;k>=2;k--)
                        {
                            mt[ar[k][0]][ar[k][1]]=1-mt[ar[k][0]][ar[k][1]];
                            indside.append((ar[k][0]+1)+" "+(ar[k][1]+1)+" ");
                        }
                        indside.append("\n");
                        ins.append(indside.toString());
                    }
                }
            }
            for(int i=n-2;i<=n-2;i++)
            {
                for(int j=0;j<=m-2;j++)
                {
                    int ar[][]=new int[4][3];
                    int c=0;

                    if(j==m-2)
                    {
                        int ones=0;
                        for(int k=0;k<=3;k++)
                        {
                            int x=i+dx[k];
                            int y=j+dy[k];
                            ar[k][0]=x;
                            ar[k][1]=y;
                            ar[k][2]=mt[x][y];
                            ones+=mt[x][y]==1?1:0;
                            //mt[x][y]=0;
                        }
                        if(ones>0) {
                            tot+=step[ones];
                            for(int k=0;k<=3;k++)
                            {
                                int x=i+dx[k];
                                int y=j+dy[k];
                                mt[x][y]=0;
                            }
                            String ret = sol(ar, ones);
                            ins.append(ret);
                        }
                    continue;
                    }
                    StringBuilder indside=new StringBuilder();
                    if(mt[i][j]==1)
                    {
                        mt[i][j]=1-mt[i][j];
                    indside.append((i+1)+" "+(j+1)+" ");
                        c++;
                    }
                    if(mt[i+1][j]==1)
                    {
                        mt[i+1][j]=1-mt[i+1][j];

                        indside.append((i+2)+" "+(j+1)+" ");
                        c++;
                    }
                    if(c>0)
                    {
                        if(c>0)
                        {
                            mt[i][j+1]=1-mt[i][j+1];

                            indside.append((i+1)+" "+(j+2)+" ");
                            c++;
                        }
                        if(c<3)
                        {
                            mt[i+1][j+1]=1-mt[i+1][j+1];
                            indside.append((i+2)+" "+(j+2)+" ");
                        }
                        tot++;
                        indside.append("\n");
                        ins.append(indside.toString());
                    }
                }
                for(int j=0;j<n;j++)
                {
                  //  System.out.println(Arrays.toString(mt[j]));
                }
            }

            for(int i=n+1;i<=n-2;i++)
            {
                for(int j=n-2;j<=m-2;j++)
                {
                    int ar[][]=new int[4][3];
                    int c=0;

                    if(i==n-2)
                    {
                        int ones=0;
                        for(int k=0;k<=3;k++)
                        {
                            int x=i+dx[k];
                            int y=j+dy[k];
                            ar[k][0]=x;
                            ar[k][1]=y;
                            ar[k][2]=mt[x][y];
                            ones+=mt[x][y]==1?1:0;
                            //mt[x][y]=0;
                        }
                        if(ones>0) {
                            tot+=step[ones];
                            for(int k=0;k<=3;k++)
                            {
                                int x=i+dx[k];
                                int y=j+dy[k];
                                mt[x][y]=0;
                            }
                            String ret = sol(ar, ones);
                            ins.append(ret);
                        }
                        continue;
                    }
                    StringBuilder indside=new StringBuilder();
                    if(mt[i][j]==1)
                    {
                        mt[i][j]=1-mt[i][j];
                        indside.append((i+1)+" "+(j+1)+" ");
                        c++;
                    }
                    if(mt[i][j+1]==1)
                    {
                        mt[i][j+1]=1-mt[i][j+1];

                        indside.append((i+1)+" "+(j+2)+" ");
                        c++;
                    }
                    if(c>0)
                    {
                        if(c>0)
                        {
                            mt[i+1][j]=1-mt[i+1][j];

                            indside.append((i+2)+" "+(j+1)+" ");
                            c++;
                        }
                        if(c<3)
                        {
                            mt[i+1][j+1]=1-mt[i+1][j+1];
                            indside.append((i+2)+" "+(j+2)+" ");
                        }
                        tot++;
                        indside.append("\n");
                        ins.append(indside.toString());
                    }
                }

                for(int j=0;j<n;j++)
                {
               //     System.out.println(Arrays.toString(mt[j]));
                }
            }

           /* for(int i=0;i<=n-2;i++)
            {
                for(int j=0;j<=m-2;j++)
                {
                    int ar[][]=new int[4][3];
                    int ones=0;
                    for(int k=0;k<=3;k++)
                    {
                        int x=i+dx[k];
                        int y=j+dy[k];
                        ar[k][0]=x;
                        ar[k][1]=y;
                        ar[k][2]=mt[x][y];
                        ones+=mt[x][y]==1?1:0;
                        //mt[x][y]=0;
                    }
                    if(ones==2||ones==3) {
                        tot+=step[ones];
                        for(int k=0;k<=3;k++)
                        {
                            int x=i+dx[k];
                            int y=j+dy[k];
                            mt[x][y]=0;
                        }
                        String ret = sol(ar, ones);
                        ins.append(ret);
                    }
                }
            }
            for(int i=0;i<=n-2;i++)
            {
                for(int j=0;j<=m-2;j++)
                {
                    int ar[][]=new int[4][3];
                    int ones=0;
                    for(int k=0;k<=3;k++)
                    {
                        int x=i+dx[k];
                        int y=j+dy[k];
                        ar[k][0]=x;
                        ar[k][1]=y;
                        ar[k][2]=mt[x][y];
                        ones+=mt[x][y]==1?1:0;
                        mt[x][y]=0;
                    }
                    if(ones==1||ones==2||ones==3) {
                        tot+=step[ones];
                        for(int k=0;k<=3;k++)
                        {
                            int x=i+dx[k];
                            int y=j+dy[k];
                            mt[x][y]=0;
                        }
                        String ret = sol(ar, ones);
                        ins.append(ret);
                    }
                }
            }
            for(int i=0;i<=n-2;i++)
            {
                for(int j=0;j<=m-2;j++)
                {
                    int ar[][]=new int[4][3];
                    int ones=0;
                    for(int k=0;k<=3;k++)
                    {
                        int x=i+dx[k];
                        int y=j+dy[k];
                        ar[k][0]=x;
                        ar[k][1]=y;
                        ar[k][2]=mt[x][y];
                        ones+=mt[x][y]==1?1:0;
                        mt[x][y]=0;
                    }
                    if(ones!=0) {
                        tot+=step[ones];
                        for(int k=0;k<=3;k++)
                        {
                            int x=i+dx[k];
                            int y=j+dy[k];
                            mt[x][y]=0;
                        }
                        String ret = sol(ar, ones);
                        ins.append(ret);
                    }
                }
            }

*/

            //System.out.println();
        sb2.append(tot+"\n");
            sb2.append(ins.toString());
        }
        System.out.println(sb2.toString());
        out.flush();
    }

    private static String sol(int[][] ar,int ones) {
    if(ones==2)
    {
        StringBuilder sb=new StringBuilder();
        int zero=0;
        for(int i=0;i<=3;i++)
        {
            if(ar[i][2]==1&&ones==2)
            {
                ones--;
            }
            else
            {
                ar[i][2]=1-ar[i][2];
                sb.append((ar[i][0]+1)+" "+(ar[i][1]+1)+" ");
            }
        }
        sb.append("\n");
        for(int i=0;i<=3;i++)
        {
            if(ar[i][2]==0)
            {
            }
            else
            {
                ar[i][2]=1-ar[i][2];
                sb.append((ar[i][0]+1)+" "+(ar[i][1]+1)+" ");
            }
        }
        sb.append("\n");
        return  sb.toString();
    }
    else if(ones==1)
    {
        StringBuilder sb=new StringBuilder();
        int zero=0;
        for(int i=0;i<=3;i++)
        {
            if(ar[i][2]==0&&zero==0)
            {
                zero++;
            }
            else
            {
                ar[i][2]=1-ar[i][2];
                sb.append((ar[i][0]+1)+" "+(ar[i][1]+1)+" ");
            }
        }
        sb.append("\n");
        sb.append(sol(ar,2));
        return  sb.toString();
    }
    else if(ones==4)
    {

        StringBuilder sb=new StringBuilder();
        int zero=0;
        for(int i=0;i<=3;i++)
        {
            if(ar[i][2]==1&&zero<3)
            {
                zero++;
                ar[i][2]=1-ar[i][2];
                sb.append((ar[i][0]+1)+" "+(ar[i][1]+1)+" ");
            }
        }
        sb.append("\n");
        sb.append(sol(ar,1));
    return  sb.toString();
    }
    else
    {
        StringBuilder sb=new StringBuilder();
        int zero=0;
        for(int i=0;i<=3;i++)
        {
            if(ar[i][2]==1&&zero<3)
            {
                zero++;
                ar[i][2]=1-ar[i][2];
                sb.append((ar[i][0]+1)+" "+(ar[i][1]+1)+" ");
            }
        }
        sb.append("\n");
    return sb.toString();
    }
    }

    public static long gethash(long pow[], long hash[], int l, int r) {

        return (hash[r+1] - (hash[l ] * pow[r - l +1]) % m + m) % m;
    }
    private static String long_pre(char[] ch) {
        int n=ch.length;
        char cpy[]=new char[n];
        for(int i=n-1;i>=0;i--)
        {
            cpy[n-1-i]=ch[i];
        }
        long pow[] = new long[n + 1];
        long hash[] = new long[n + 1];
        pow[0] = 1;
        int p = 31;
        for (int i = 1; i <= n; i++) {
            pow[i] = (pow[i - 1] % m * p % m) % m;
        }

        for (int i = 1; i <= n; i++) {
            hash[i] = (hash[i - 1] * p % m + (ch[i - 1]) % m) % m;
        }

        long pow1[] = new long[n + 1];
        long hash1[] = new long[n + 1];
        pow1[0] = 1;
        //int p = 31;
        for (int i = 1; i <= n; i++) {
            pow1[i] = (pow1[i - 1] % m * p % m) % m;
        }

        for (int i = 1; i <= n; i++) {
            hash1[i] = (hash1[i - 1] * p % m + (cpy[i - 1]) % m) % m;
        }


        int l=0,r=n-1,lst=0;
        for(int i=n-1;i>=1;i--)
        {
        int mid=i%2;
        int midp=i/2;
        if(mid==0&&gethash(pow,hash,0,midp-1)==gethash(pow1,hash1,n-1-i,n-1-(midp+1)))
        {
            lst=i;
            break;
        }
        else if(mid==1&&gethash(pow,hash,0,midp)==gethash(pow1,hash1,n-1-i,n-1-(midp+1)))
        {
            lst=i;
            break;
        }
        }
        StringBuilder sb=new StringBuilder();
        for(int i=0;i<=lst;i++)
        {
            sb.append(ch[i]);
        }
        return sb.toString();
    }

    private static int Solve(int n, int l, int r,int h, int i, int s, int[] ar) {
        if(i==n)
            return 0;
        if(dp[i][s]!=-1)
            return dp[i][s];

        int ans=Math.max(con((s+ar[i])%h,l,r)+Solve(n,l,r, h,i+1,(s+ar[i])%h,ar)
                ,con((s+ar[i]-1)%h,l,r)+Solve(n,l,r,h,i+1,(s+ar[i]-1+h)%h,ar));
        dp[i][s]=ans;
        return dp[i][s];
    }

    private static int con(int i,int l,int r) {
        if(l<=i&&i<=r)
            return 1;
        else
            return 0;
    }

    static  long childs[]=new long[max];
    static long tot=0;
    public static void dfs_val(List<Integer> adj[],int u,int p,long val) {
        int in_max=-1;
        val=val+childs[u]+1;
        for(int v:adj[u])
        {
            if(v!=p)
            {
                in_max=v;
                dfs_val(adj,v,p,val);
            }
        }
        if(in_max==-1)
        {
            tot=Math.max(tot,val);
        }
    }
    public static void dfs(List<Integer> adj[],int u,int p)
    {

        for(int v:adj[u])
        {
            if(v!=p)
            {
                childs[u]++;
                dfs(adj,v,u);
                childs[u]+=childs[v];
            }
        }
    }
    public static int ans(int ar[],int i,int n,int t)
    {
        if(i>=n)
            return 0;
        if(t>=410)
            return 200000;
        if(dp[i][t]!=-1)
            return dp[i][t];
        //System.out.println(t+" "+i);
        int cur=Math.min(Math.abs(ar[i]-t)+ans(ar,i+1,n,t+1),ans(ar,i,n,t+1));
        dp[i][t]=cur;
        return dp[i][t];
    }

    private static long ncr(int n, int r, long[] fact) {
        if(r>n||n < 0 || r < 0)
            return 0;

        if(n==1||n==0)
            return 1;
        // System.out.println(n+" c "+r);

        return (((fact[n]*pow_inverse(fact[n-r],p))%p)*pow_inverse(fact[r],p))%p;
    }
    static class BIT {
        long[] tree;
        public BIT(int size) {
            tree = new long[size + 1];
        }
        public long sum(int i) {
            long ans = 0;
            while (i > 0) {
                ans += tree[i];
                i -= i & -i;
            }
            return ans;
        }

        public long query(int i, int j) {
            return sum(j) - sum(i - 1);
        }

        public void add(int i, long val) {
            while (i < tree.length) {
                tree[i] += val;
                i += i & -i;
            }
        }

        public void set(int i, long val) {
            add(i, val - query(i, i));
        }
    }
    static class  vert implements Comparator<vert>
    {
        int v;
        TreeSet<Integer> set;

        public vert(int v, TreeSet<Integer> set) {
            this.v = v;
            this.set = set;
        }

        @Override
        public int compare(vert o1, vert o2) {
            return -(o1.set.size()-o2.set.size());
        }
    }
    public static long pow(long a,long b,long m)
    {
        long r=1;
        while(b!=1)
        {
            if(b%2!=0)
                r=(r*a)%m;
            b=b>>1;
            a=(a*a)%m;
        }
        return (r*a)%m;
    }
    public static long pow_inverse(long a,long m)
    {
        return pow(a,m-2,m);
    }public static long gcd(long a,long b)
    {
        if(b>a)
        {
            a=a^b;
            b=a^b;
            a=a^b;
        }
        if(b==0)
            return a;
        return gcd(b,a%b);
    }
    public static void extgcd(long a,long b)
    {/*
            if(a==0)
            {
                x=0;
                y=1;
                d=b;
                // return ;
            }
            else
            {

                extgcd(b%a,a);

                long t=y;
                y=x;
                x=t-(b/a)*x;
            }*/
    }

    public static void seive()
    {
        for(int i=0;i<max;i++)
            prime[i] = true;

        for(int p = 2; p*p <=max; p++)
        {
            // If prime[p] is not changed, then it is a prime
            if(prime[p] == true)
            {
                // Update all multiples of p
                for(int i = p*p; i <=max; i += p)
                    prime[i] = false;
            }
        }

        // Print all prime numbers
        for(int i = 2; i <= max; i++)
        {
            if(prime[i] == true)
            {
                total_prime.add(i);
            }
        }
    }
    public static boolean union(int a, int b, int p[], int r[]){
        // add your code here
        int pa=find(a,p,r);
        int pb=find(b,p,r);
        if(pa==pb)
            return true;
        if(pa!=pb)
        {
            if(r[pa]<r[pb])
            {
                p[pa]=pb;
            }
            else if(r[pa]>r[pb])
            {
                p[pb]=pa;
            }
            else
            {
                p[pa]=pb;
                r[pb]++;
            }
        }
        return false;
    }
    public static int find(int x,int p[],int r[])
    {
        if(p[x]==x)
            return x;
        p[x]=find(p[x],p,r);
        return p[x];
    }
    public static int binary_Search_upper(Integer[] ar, Integer x,int l,int r)
    {
        int res=-1,f=0;
        while(l<=r)
        {
            int mid=(l+r)>>1;
            if(ar[mid]==x)
            {
                f=1;
                res=mid;
                l=mid+1;
            }
            else if(ar[mid]>x)
            {
                res=res!=-1&&ar[res]==x?res:mid;
                r=mid-1;
            }
            else
            {
                l=mid+1;
            }
        }
        return f==1?res:res+1;
    }
    public static int binary_Search_lower(int[] ar, long x, int l, int r)
    {
        int res=0,f=0;
        //int l=0;int r=ar.length-1;
        /*public static int binary(Long[] ar, int l, int r, long v)
        {
            int n=r;
            int m=(l+r)/2;
            int re=-1;
            while(l<=r)
            {
                if(ar[m]>=v)
                {

                    re=m;
                    r=m-1;
                }
                else if(ar[m]<v)
                {
                    l=m+1;
                }
                m=(l+r)/2;
            }
            if(l<=n&&ar[l]==v)
                return l;
            return re;
        }*/
        //   System.out.println(Arrays.toString(ar)+" "+x);
        while(l<=r)
        {
            int mid=(l+r)>>1;
            //   System.out.println(l+" "+r+" "+mid+" = "+ar[mid]);
            if(ar[mid]==x)
            {
                f=1;
                res=mid;
                r=mid-1;
            }
            else if(ar[mid]>x)
            {
                r=mid-1;
            }
            else
            {
                res=res!=0&&ar[res]==x?res:mid;
                l=mid+1;
            }
        }

        //  System.out.println(res);
        return f==1?res:res+1;
    } public static int binary_Search_lower_suf(int[] ar, int x,int l,int r)
    {
        int res=-1,f=0;
        //int l=0;int r=ar.length-1;
        while(l<=r)
        {
            int mid=(l+r)>>1;
            if(ar[mid]==x)
            {
                f=1;
                res=mid;
                r=mid-1;
            }
            else if(ar[mid]<x)
            {
                r=mid-1;
            }
            else
            {
                //      res=mid;
                l=mid+1;
            }
        }
        return f==0?res+1:res;
    }
    int bit[]=new int[(int)1e6];
    public void update(int n,int val,int i)
    {
        i++;
        while(i<n)
        {
            bit[i]+=val;
            i+=(i)&(-i);
        }

    }
    public long query(int n,int i)
    {
        i++;
        long sum=0;
        while(i>0)
        {
            sum+=bit[i];
            i-=(i)&(-i);
        }
        return sum;
    }


    ///////////////////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////////////////////////////
    static class FastReader {

        BufferedReader br;
        StringTokenizer st;

        public FastReader() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }

        String next() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }

        String nextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }
    }


}