I would use the
our problem can be approached using a backtracking algorithm. You may try each list from the first list, subtract it from the second list (if possible), and then recursively try to fit the next list from the first list into the remainder of the second list. The goal is to find the maximum number of lists that can be subtracted in this manner.

I would approach it like following:

def can_subtract(list1, list2):
    """
    Try to subtract list1 from list2 starting from any position.
    Returns the resulting list if possible, otherwise None.
    """
    n, m = len(list1), len(list2)
    for i in range(m - n + 1):  # Try all starting positions in list2
        if all(list2[i+j] >= list1[j] for j in range(n)):
            # Subtract list1 from list2 starting from index i
            new_list2 = list2[:]
            for j in range(n):
                new_list2[i+j] -= list1[j]
            return new_list2
    return None

def max_lists_subtracted(lists, target_list):
    def backtrack(index, current_list, count):
        nonlocal max_count
        if index >= len(lists):
            max_count = max(max_count, count)
            return
        
        # Try to subtract the current list
        result = can_subtract(lists[index], current_list)
        if result is not None:
            # If successful, move to the next list
            backtrack(index + 1, result, count + 1)
        
        # Also consider not using the current list and moving to the next
        backtrack(index + 1, current_list, count)
    
    max_count = 0
    backtrack(0, target_list, 0)
    return max_count

# Example usage:
lists = [[4, 4], [4, 5, 6], [8, 9, 1]]
target_list = [8, 9, 7, 1]
print(max_lists_subtracted(lists, target_list))  # Output is 2 

So, here

1. can_subtract(list1, list2): Checks if list1 can be subtracted from list2 starting at any position. If possible, it returns the modified list2 after subtraction; otherwise, it returns None.
2. Secondly, max_lists_subtracted(lists, target_list) uses backtracking to try subtracting each list in lists from the target_list. It keeps track of the maximum number of lists that can be subtracted.