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 PYTHON3):

Solution:
#!/usr/bin/env python
from __future__ import division, print_function

import os
import sys
from io import BytesIO, IOBase

if sys.version_info[0] < 3:
    from __builtin__ import xrange as range
    from future_builtins import ascii, filter, hex, map, oct, zip


def main():
    t = int(input())

    def make_op(table, op):
        table[op[0]][op[1]] = 1 - table[op[0]][op[1]]
        table[op[2]][op[3]] = 1 - table[op[2]][op[3]]
        table[op[4]][op[5]] = 1 - table[op[4]][op[5]]

    for _ in range(t):
        n, m = map(int, input().split())
        table = [list(map(int, input())) for _ in range(n)]

        ops = []

        for i in range(0, n - 1, 2):
            for j in range(m - 1):
                if table[i][j] and table[i + 1][j]:
                    op = [i, j, i + 1, j, i, j + 1]
                    ops.append(op)
                    make_op(table, op)
                elif table[i][j]:
                    op = [i, j, i, j + 1, i + 1, j + 1]
                    ops.append(op)
                    make_op(table, op)
                elif table[i + 1][j]:
                    op = [i + 1, j, i, j + 1, i + 1, j + 1]
                    ops.append(op)
                    make_op(table, op)

        i = n - 2
        for j in range(m - 1):
            if table[i][j] and table[i + 1][j]:
                op = [i, j, i + 1, j, i, j + 1]
                ops.append(op)
                make_op(table, op)
            elif table[i][j]:
                op = [i, j, i, j + 1, i + 1, j + 1]
                ops.append(op)
                make_op(table, op)
            elif table[i + 1][j]:
                op = [i + 1, j, i, j + 1, i + 1, j + 1]
                ops.append(op)
                make_op(table, op)

        j = m - 2
        for i in range(n - 2):
            if table[i][j] and table[i][j + 1]:
                op = [i, j, i, j + 1, i + 1, j]
                ops.append(op)
                make_op(table, op)
            elif table[i][j]:
                op = [i, j, i + 1, j, i + 1, j + 1]
                ops.append(op)
                make_op(table, op)
            elif table[i][j + 1]:
                op = [i, j + 1, i + 1, j, i + 1, j + 1]
                ops.append(op)
                make_op(table, op)

        while True:
            ones = []
            zeros = []
            for i in range(n - 2, n):
                for j in range(m - 2, m):
                    if table[i][j]: ones.append([i, j])
                    else: zeros.append([i, j])

            if len(ones) == 0:
                break
            elif len(ones) <= 2:
                op = [*ones.pop(), *zeros.pop(), *zeros.pop()]
                ops.append(op)
                make_op(table, op)
            elif len(ones) >= 3:
                op = [*ones.pop(), *ones.pop(), *ones.pop()]
                ops.append(op)
                make_op(table, op)

        assert len(ops) <= n * m
        print(len(ops))
        for op in ops:
            print(*[i + 1 for i in op])

        #for row in table:
        #    print(*row, sep='')


# region fastio

BUFSIZE = 8192


class FastIO(IOBase):
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def print(*args, **kwargs):
    """Prints the values to a stream, or to sys.stdout by default."""
    sep, file = kwargs.pop("sep", " "), kwargs.pop("file", sys.stdout)
    at_start = True
    for x in args:
        if not at_start:
            file.write(sep)
        file.write(str(x))
        at_start = False
    file.write(kwargs.pop("end", "\n"))
    if kwargs.pop("flush", False):
        file.flush()


if sys.version_info[0] < 3:
    sys.stdin, sys.stdout = FastIO(sys.stdin), FastIO(sys.stdout)
else:
    sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)

input = lambda: sys.stdin.readline().rstrip("\r\n")

# endregion

if __name__ == "__main__":
    main()