def sum_scales(data: list[list[tuple[int, int, int, int, int]]]):
    """Calculate the sum of scales for each axis across multiple datasets."""    
    if not data or not data[0]:
        raise ValueError("Input data must be a non-empty list of non-empty lists.")
    
        
    result = []
    for i, base_item in enumerate(data[0]):
        val1, val2, val3, base_scale1, base_scale2 = base_item
        summed_scale1 = base_scale1
        summed_scale2 = base_scale2
        
        # Sum corresponding scales from all other datasets
        for dataset_idx in range(1, len(data)):
            if i < len(data[dataset_idx]):  # Check if index exists in this dataset
                _, _, _, scale1, scale2 = data[dataset_idx][i]
                summed_scale1 += scale1
                summed_scale2 += scale2
        
        # Create new tuple with summed scales
        result.append((val1, val2, val3, summed_scale1, summed_scale2))
        
    return result

def difference_scales(data1: list[tuple[int, int, int, int, int]], 
                      data2: list[tuple[int, int, int, int, int]]) -> list[tuple[int, int, int, int, int]]:
    """
    Calculate the difference between scales of two datasets.
    
    For each pair of tuples, calculates the square root of the squared difference
    between corresponding scales, divided by 2.
    
    Args:
        data1: First dataset with 5-tuple elements
        data2: Second dataset with 5-tuple elements
        
    Returns:
        List of tuples with differences between scales
    """
    import math
    result = []
    
    # Use the shorter dataset length to avoid index errors
    min_length = min(len(data1), len(data2))
    
    for i in range(min_length):
        val1_1, val1_2, val1_3, scale1_1, scale1_2 = data1[i]
        val2_1, val2_2, val2_3, scale2_1, scale2_2 = data2[i]
        
        # Calculate scale differences using the formula
        diff_scale1 = math.sqrt((scale1_1 - scale2_1) ** 2) / 2
        diff_scale2 = math.sqrt((scale1_2 - scale2_2) ** 2) / 2
        
        # Create new tuple: keeping val2_1, val2_2, val2_3 from second dataset
        # and including the calculated scale differences
        result.append((val2_1, val2_2, val2_3, diff_scale1, diff_scale2))
    
    return result

def difference_values(data1: list[tuple[int, int, int, int, int]], 
                      data2: list[tuple[int, int, int, int, int]]) -> list[tuple[int, int, int, int, int]]:
    """
    Calculate the difference between values of two datasets. (as scales)
    
    For each pair of tuples, calculates the square root of the squared difference
    between corresponding values.
    
    Args:
        data1: First dataset with 5-tuple elements
        data2: Second dataset with 5-tuple elements
        
    Returns:
        List of tuples with differences between values
    """
    import math
    result = []
    
    # Use the shorter dataset length to avoid index errors
    min_length = min(len(data1), len(data2))
    
    for i in range(min_length):
        val1_1, val1_2, val1_3, scale1_1, scale1_2 = data1[i]
        val2_1, val2_2, val2_3, scale2_1, scale2_2 = data2[i]
        
        # Calculate scale differences using the formula
        diff_scale_tmp = val1_3 - val2_3
        
        diff_scale1 = diff_scale_tmp if diff_scale_tmp > 0 else 0
        diff_scale2 = diff_scale_tmp if diff_scale_tmp < 0 else 0
        
        # Create new tuple: keeping val2_1, val2_2, val2_3 from second dataset
        # and including the calculated scale differences
        result.append((val2_1, val2_2, val2_3, diff_scale1, diff_scale2))
    
    return result

def sum_quad_scales(data: list[list[tuple[int, int, int, int, int]]]):
    """Calculate the sum of scales for each axis across multiple datasets."""    
    from math import sqrt
    if not data or not data[0]:
        raise ValueError("Input data must be a non-empty list of non-empty lists.")
    
        
    result = []
    for i, base_item in enumerate(data[0]):
        val1, val2, val3, base_scale1, base_scale2 = base_item
        summed_scale1 = base_scale1**2
        summed_scale2 = base_scale2**2
        
        # Sum corresponding scales from all other datasets
        for dataset_idx in range(1, len(data)):
            if i < len(data[dataset_idx]):  # Check if index exists in this dataset
                _, _, _, scale1, scale2 = data[dataset_idx][i]
                summed_scale1 += scale1**2
                summed_scale2 += scale2**2
        
        # Create new tuple with summed scales
        result.append((val1, val2, val3, sqrt(summed_scale1), sqrt(summed_scale2)))
        
    return result

def two_scales_envelope(data1: list[tuple[int, int, int, int, int]], 
                      data2: list[tuple[int, int, int, int, int]]) -> list[tuple[int, int, int, int, int]]:
    """
    Calculate the envelope between two datasets. (as scales)
    
    For each pair of tuples, calculates max of each scale.
    
    Args:
        data1: First dataset with 5-tuple elements
        data2: Second dataset with 5-tuple elements
        
    Returns:
        List of tuples with envelope between values
    """
    import math
    result = []
    
    # Use the shorter dataset length to avoid index errors
    min_length = min(len(data1), len(data2))
    
    for i in range(min_length): 
        val1_1, val1_2, val1_3, scale1_1, scale1_2 = data1[i]
        val2_1, val2_2, val2_3, scale2_1, scale2_2 = data2[i]
        assert val1_3 == val2_3, "Values must be equal for envelope calculation"
        
        diff_scale1 = max(scale1_1, scale2_1)
        diff_scale2 = max(scale1_2, scale2_2)
        
        # Create new tuple: keeping val2_1, val2_2, val2_3 from second dataset
        # and including the calculated scale differences
        result.append((val2_1, val2_2, val2_3, diff_scale1, diff_scale2))
    
    return result