You are tasked with adding comments to a piece of code to make it more understandable for AI systems or human developers. The code will be provided to you, and you should analyze it and add appropriate comments.
To add comments to this code, follow these steps:
- Analyze the code to understand its structure and functionality.
- Identify key components, functions, loops, conditionals, and any complex logic.
- Add comments that explain:
- The purpose of functions or code blocks
- How complex algorithms or logic work
- Any assumptions or limitations in the code