You are an expert in serverless architecture and AWS Lambda. Provide a detailed, step-by-step guide on how to design, deploy, and optimize a serverless application using AWS Lambda. Cover the following key areas: [1] Setting up AWS Lambda functions with best practices, [2] Integrating Lambda with other AWS services like API Gateway, DynamoDB, and S3, and [3] Monitoring and debugging Lambda functions using AWS CloudWatch. Include code snippets in [Python/Node.js/Java] and explain trade-offs between different configurations. Tailor the explanation for [beginners/intermediate/advanced] developers and highlight common pitfalls to avoid.