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def add_lists_index_wise(list1, list2):
    # Use zip() to combine the two lists
    combined = zip(list1, list2)
    
    # Initialize an empty list to store the results
    result = []
    
    # Iterate over the combined zip object
    for items in combined:
        # Add the elements together as strings and append to the result list
        result.append(''.join(items))
    
    # If the input lists are of different lengths, append the remaining items
    result.extend(list1[len(list2):])
    result.extend(list2[len(list1):])
    
    return result

# Test the function
list1 = ["M", "na", "i", "Ke"]
list2 = ["y", "me", "s", "lly"]
print(add_lists_index_wise(list1, list2))  # Output: ['My', 'name', 'is', 'Kelly']
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