Autocomplete System

Design and implement an autocomplete system. Your system should store a dynamic dictionary of words and allow efficient retrieval of all words that match a given prefix. The system must support the following operations:

1. insert(word): Add a new word to the dictionary.
2. delete(word): Remove an existing word from the dictionary.
3. search(prefix): Return a list of all words in the dictionary that start with the given prefix. The results should be returned in lexicographical order.

Requirements:

• The dictionary should handle a large number of words efficiently.
• The operations should be optimized for performance, especially the prefix search.
• You may choose any programming language and data structures that you feel are appropriate, but you should justify your choices if necessary in comments.

Consider both time and space complexities, and ensure that your solution can handle dynamic updates to the dictionary.

Provide a complete, self-contained implementation that includes examples demonstrating how your system works.