Build a basic shopping cart system that can handle a few functional requirements. Implement process_cart(prices, operations), where prices maps item names to integer prices and operations applies add, update, or remove actions. Return {"items": ..., "total": ...}, with items ordered lexicographically and total calculated from the final quantities. Add increases quantity, update replaces it, update with zero removes the item, and remove deletes it.
def process_cart(prices, operations):